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# Number System Notes | EduRev

## : Number System Notes | EduRev

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Number'System'
The\$numbers\$can\$be\$defined\$in\$a\$lot\$of\$different\$ways\$like\$positive,\$negative,\$even,\$
odd,\$natural,\$whole,\$integers,\$fractions,\$etc.\$
This\$chapter\$deals\$with\$all\$these\$i.e\$different\$kinds\$of\$numbers.\$
\$
Positive/Negative'
Numbers\$can\$either\$be\$positive\$or\$negative\$or\$even\$none\$of\$the\$these!!\$
\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$
\$ >5\$ >4\$ >3\$ >2\$ >1\$ 0\$ 1\$ 2\$ 3\$ 4\$ 5\$
A\$number\$line\$illustrates\$this.\$
A\$negative\$number\$is\$defined\$as\$any\$number\$left\$to\$the\$zero\$or\$a\$number\$less\$than\$
zero.\$The\$symbol\$used\$to\$denote\$a\$negative\$number\$is\$-.\$
A\$positive\$number\$is\$defined\$as\$any\$number\$right\$to\$the\$zero\$or\$a\$number\$more\$than\$
zero.\$The\$symbol\$used\$to\$denote\$a\$positive\$number\$is\$+.\$
As\$we\$can\$see\$that\$the\$number\$zero\$is\$just\$used\$to\$make\$a\$distinction\$between\$
positive\$and\$negative\$number,\$so\$it\$is\$considered\$to\$be\$neither\$positive\$or\$negative\$
i.e\$zero'(0)'is'a'neutral'number.'
Things'to'keep'in'mind:0'
1. Positive\$×\$positive\$=\$positive\$
2. positive\$\$×\$negative\$=\$negative\$\$\$\$\$\$Multiplication\$
3. negative\$\$×\$negative\$=\$positive\$
4. positive/positive\$=\$positive\$\$\$\$\$\$\$\$\$\$\$
5. positive/negative\$=\$negative\$\$\$\$\$\$\$\$\$\$Division\$
6. negative/negative\$=\$positive\$
7. A\$double\$negative\$means\$positive.\$For\$example:>\$4\$–\$(>2)\$=\$4\$+\$2=\$6\$
A"few"more"definitions:>\$
1. Natural\$numbers:>\$1,\$2,\$3,\$4,\$5,\$………..(only\$positive)\$
2. Whole\$numbers:>\$0,\$1,\$2,\$3,\$………………(non\$–\$negative)\$
3. Integers:>……………….\$>5,\$>4,\$>3,\$>2,\$>1,\$0,\$1,\$2,\$3……………(negative/positive)\$
4. Fractions:>\$numbers\$of\$the\$form\$p/q\$where\$q\$?\$0\$
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Number'System'
The\$numbers\$can\$be\$defined\$in\$a\$lot\$of\$different\$ways\$like\$positive,\$negative,\$even,\$
odd,\$natural,\$whole,\$integers,\$fractions,\$etc.\$
This\$chapter\$deals\$with\$all\$these\$i.e\$different\$kinds\$of\$numbers.\$
\$
Positive/Negative'
Numbers\$can\$either\$be\$positive\$or\$negative\$or\$even\$none\$of\$the\$these!!\$
\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$
\$ >5\$ >4\$ >3\$ >2\$ >1\$ 0\$ 1\$ 2\$ 3\$ 4\$ 5\$
A\$number\$line\$illustrates\$this.\$
A\$negative\$number\$is\$defined\$as\$any\$number\$left\$to\$the\$zero\$or\$a\$number\$less\$than\$
zero.\$The\$symbol\$used\$to\$denote\$a\$negative\$number\$is\$-.\$
A\$positive\$number\$is\$defined\$as\$any\$number\$right\$to\$the\$zero\$or\$a\$number\$more\$than\$
zero.\$The\$symbol\$used\$to\$denote\$a\$positive\$number\$is\$+.\$
As\$we\$can\$see\$that\$the\$number\$zero\$is\$just\$used\$to\$make\$a\$distinction\$between\$
positive\$and\$negative\$number,\$so\$it\$is\$considered\$to\$be\$neither\$positive\$or\$negative\$
i.e\$zero'(0)'is'a'neutral'number.'
Things'to'keep'in'mind:0'
1. Positive\$×\$positive\$=\$positive\$
2. positive\$\$×\$negative\$=\$negative\$\$\$\$\$\$Multiplication\$
3. negative\$\$×\$negative\$=\$positive\$
4. positive/positive\$=\$positive\$\$\$\$\$\$\$\$\$\$\$
5. positive/negative\$=\$negative\$\$\$\$\$\$\$\$\$\$Division\$
6. negative/negative\$=\$positive\$
7. A\$double\$negative\$means\$positive.\$For\$example:>\$4\$–\$(>2)\$=\$4\$+\$2=\$6\$
A"few"more"definitions:>\$
1. Natural\$numbers:>\$1,\$2,\$3,\$4,\$5,\$………..(only\$positive)\$
2. Whole\$numbers:>\$0,\$1,\$2,\$3,\$………………(non\$–\$negative)\$
3. Integers:>……………….\$>5,\$>4,\$>3,\$>2,\$>1,\$0,\$1,\$2,\$3……………(negative/positive)\$
4. Fractions:>\$numbers\$of\$the\$form\$p/q\$where\$q\$?\$0\$
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Even/Odd'
Even:>\$An\$even\$number\$is\$an\$integer\$that\$is\$divisible\$by\$2.\$For\$example:>\$>24,\$>36,\$>20,\$
0,\$20,\$42,\$38\$etc.\$\$\$An\$even\$number\$can\$be\$written\$in\$the\$form\$n\$=\$2k,\$where\$k\$is\$an\$
integer.\$
Odd:>\$An\$odd\$number\$is\$an\$integer\$that\$is\$not\$divisible\$by\$2.\$For\$example:>\$>23,\$>37,\$>
19,\$1,\$3,\$17,\$etc.\$\$\$\$An\$odd\$number\$can\$be\$written\$in\$the\$form\$n\$=\$2k+1,\$where\$k\$is\$an\$
integer.\$
Things'to'remember:0\$
1. even\$+\$even\$=\$even\$
3. odd\$+\$odd\$=\$even\$
4. even\$–\$even\$=\$even\$
5. even\$–\$odd\$=\$odd\$\$\$\$\$\$\$\$\$\$\$\$Subtraction\$
6. odd\$–\$odd\$=\$even\$
7. even\$×\$even\$=\$even\$
8. even\$×\$odd\$=\$even\$\$\$\$\$\$\$\$\$Multiplication\$
9. odd\$×\$odd\$=\$odd\$
10. Division\$of\$even\$or\$odd\$numbers\$does\$not\$follow\$any\$specific\$rules.\$It\$may\$
result\$in\$an\$even\$or\$odd\$integer\$or\$a\$fraction.\$For\$example:\$>\$6/2\$=\$3,\$6/3\$=\$2,\$
35/5\$=\$7,\$6/4\$?\$integer\$
11. The\$only\$specific\$rule\$for\$division\$is\$(Odd/Even)\$?\$integer\$i.e\$an\$odd\$integer\$
when\$divided\$by\$an\$even\$integer\$would\$never\$result\$in\$an\$integer.\$
\$
Consecutive'Integers:A'
The\$word\$consecutive\$means\$one\$after\$the\$other.\$Similarly,\$consecutive\$numbers\$are\$
the\$numbers\$that\$follow\$one\$another\$from\$a\$given\$value.\$
For\$Example:>\$1,\$2,\$3,\$4\$are\$consecutive\$integers\$and\$\$\$\$\$\$\$\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$>12,\$>13,\$>14,\$>15,\$16\$are\$also\$consecutive\$integers.\$\$\$\$
Consecutive\$integers\$can\$also\$make\$some\$specific\$patterns\$like:>\$
• Consecutive\$even\$integers:>\$2,\$4,\$6,\$8,\$10………\$
• Consecutive\$odd\$integers:>\$1,\$3,\$5,\$7,\$9……\$
• Consecutive\$multiples\$of\$5:>\$5,\$10,\$15,\$20,\$25………..\$
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Number'System'
The\$numbers\$can\$be\$defined\$in\$a\$lot\$of\$different\$ways\$like\$positive,\$negative,\$even,\$
odd,\$natural,\$whole,\$integers,\$fractions,\$etc.\$
This\$chapter\$deals\$with\$all\$these\$i.e\$different\$kinds\$of\$numbers.\$
\$
Positive/Negative'
Numbers\$can\$either\$be\$positive\$or\$negative\$or\$even\$none\$of\$the\$these!!\$
\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$
\$ >5\$ >4\$ >3\$ >2\$ >1\$ 0\$ 1\$ 2\$ 3\$ 4\$ 5\$
A\$number\$line\$illustrates\$this.\$
A\$negative\$number\$is\$defined\$as\$any\$number\$left\$to\$the\$zero\$or\$a\$number\$less\$than\$
zero.\$The\$symbol\$used\$to\$denote\$a\$negative\$number\$is\$-.\$
A\$positive\$number\$is\$defined\$as\$any\$number\$right\$to\$the\$zero\$or\$a\$number\$more\$than\$
zero.\$The\$symbol\$used\$to\$denote\$a\$positive\$number\$is\$+.\$
As\$we\$can\$see\$that\$the\$number\$zero\$is\$just\$used\$to\$make\$a\$distinction\$between\$
positive\$and\$negative\$number,\$so\$it\$is\$considered\$to\$be\$neither\$positive\$or\$negative\$
i.e\$zero'(0)'is'a'neutral'number.'
Things'to'keep'in'mind:0'
1. Positive\$×\$positive\$=\$positive\$
2. positive\$\$×\$negative\$=\$negative\$\$\$\$\$\$Multiplication\$
3. negative\$\$×\$negative\$=\$positive\$
4. positive/positive\$=\$positive\$\$\$\$\$\$\$\$\$\$\$
5. positive/negative\$=\$negative\$\$\$\$\$\$\$\$\$\$Division\$
6. negative/negative\$=\$positive\$
7. A\$double\$negative\$means\$positive.\$For\$example:>\$4\$–\$(>2)\$=\$4\$+\$2=\$6\$
A"few"more"definitions:>\$
1. Natural\$numbers:>\$1,\$2,\$3,\$4,\$5,\$………..(only\$positive)\$
2. Whole\$numbers:>\$0,\$1,\$2,\$3,\$………………(non\$–\$negative)\$
3. Integers:>……………….\$>5,\$>4,\$>3,\$>2,\$>1,\$0,\$1,\$2,\$3……………(negative/positive)\$
4. Fractions:>\$numbers\$of\$the\$form\$p/q\$where\$q\$?\$0\$
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Even/Odd'
Even:>\$An\$even\$number\$is\$an\$integer\$that\$is\$divisible\$by\$2.\$For\$example:>\$>24,\$>36,\$>20,\$
0,\$20,\$42,\$38\$etc.\$\$\$An\$even\$number\$can\$be\$written\$in\$the\$form\$n\$=\$2k,\$where\$k\$is\$an\$
integer.\$
Odd:>\$An\$odd\$number\$is\$an\$integer\$that\$is\$not\$divisible\$by\$2.\$For\$example:>\$>23,\$>37,\$>
19,\$1,\$3,\$17,\$etc.\$\$\$\$An\$odd\$number\$can\$be\$written\$in\$the\$form\$n\$=\$2k+1,\$where\$k\$is\$an\$
integer.\$
Things'to'remember:0\$
1. even\$+\$even\$=\$even\$
3. odd\$+\$odd\$=\$even\$
4. even\$–\$even\$=\$even\$
5. even\$–\$odd\$=\$odd\$\$\$\$\$\$\$\$\$\$\$\$Subtraction\$
6. odd\$–\$odd\$=\$even\$
7. even\$×\$even\$=\$even\$
8. even\$×\$odd\$=\$even\$\$\$\$\$\$\$\$\$Multiplication\$
9. odd\$×\$odd\$=\$odd\$
10. Division\$of\$even\$or\$odd\$numbers\$does\$not\$follow\$any\$specific\$rules.\$It\$may\$
result\$in\$an\$even\$or\$odd\$integer\$or\$a\$fraction.\$For\$example:\$>\$6/2\$=\$3,\$6/3\$=\$2,\$
35/5\$=\$7,\$6/4\$?\$integer\$
11. The\$only\$specific\$rule\$for\$division\$is\$(Odd/Even)\$?\$integer\$i.e\$an\$odd\$integer\$
when\$divided\$by\$an\$even\$integer\$would\$never\$result\$in\$an\$integer.\$
\$
Consecutive'Integers:A'
The\$word\$consecutive\$means\$one\$after\$the\$other.\$Similarly,\$consecutive\$numbers\$are\$
the\$numbers\$that\$follow\$one\$another\$from\$a\$given\$value.\$
For\$Example:>\$1,\$2,\$3,\$4\$are\$consecutive\$integers\$and\$\$\$\$\$\$\$\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$>12,\$>13,\$>14,\$>15,\$16\$are\$also\$consecutive\$integers.\$\$\$\$
Consecutive\$integers\$can\$also\$make\$some\$specific\$patterns\$like:>\$
• Consecutive\$even\$integers:>\$2,\$4,\$6,\$8,\$10………\$
• Consecutive\$odd\$integers:>\$1,\$3,\$5,\$7,\$9……\$
• Consecutive\$multiples\$of\$5:>\$5,\$10,\$15,\$20,\$25………..\$
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Things'to'remember:0'
1. The\$arithmetic\$mean\$(average)\$is\$equal\$to\$the\$median\$in\$a\$set\$of\$consecutive\$
numbers.\$
2. The\$average\$and\$the\$median\$are\$both\$equal\$to\$the\$average\$of\$the\$1
st
\$and\$last\$
numbers\$of\$the\$set.\$\$
For"Example:>\$In\$the\$set\$2,\$4,\$6,\$8……………..200,\$the\$average\$and\$the\$median\$are\$both\$
equal\$to\$the\$average\$of\$the\$1
st
\$and\$the\$last\$numbers\$i.e\$Average\$=\$median\$=\$
(2+200)/2=\$101\$
3. Two\$consecutive\$integers\$are\$never\$divisible\$by\$the\$same\$prime\$number\$and\$
therefore\$by\$the\$same\$number.\$\$\$
the\$result.\$
For\$example:>\$The\$number\$of\$integers\$from\$2\$to\$7\$is\$not\$5\$but\$6\$(2,\$3,\$,4,\$\$5,\$6\$
and\$7)\$
But\$the\$number\$of\$integers\$between\$2\$and\$7\$is\$5.\$
5. The\$product\$of\$n\$consecutive\$numbers\$is\$always\$divisible\$by\$n.\$
6. The\$sum\$of\$n\$consecutive\$integers\$is\$always\$divisible\$by\$n\$if\$n\$is\$odd\$and\$
never\$divisible\$by\$n\$if\$n\$is\$even.\$
For\$example:>\$The\$sum\$of\$1,\$2\$and\$3\$i.e\$6\$is\$divisible\$by\$3(number\$of\$integers\$
is\$odd)\$but\$he\$sum\$of\$1,\$2,\$3\$and\$4\$i.e\$10\$is\$not\$divisible\$by\$4(number\$of\$
integers\$is\$even)\$
Divisibility'Rules:A''
A\$few\$important\$divisibility\$rules:>\$
2\$>\$A\$number\$is\$divisible\$by\$2\$if\$the\$last\$digit\$of\$the\$number\$is\$even\$
3>\$A\$number\$is\$divisible\$by\$3\$if\$the\$sum\$of\$the\$digits\$of\$the\$number\$is\$divisible\$by\$3\$
4>\$A\$number\$is\$divisible\$by\$4\$if\$the\$last\$two\$digits\$form\$a\$number\$that\$is\$divisible\$by\$4\$
5>\$A\$number\$is\$divisible\$by\$5\$if\$the\$last\$digit\$is\$0\$or\$5\$
6>\$A\$number\$is\$divisible\$by\$6\$if\$the\$number\$is\$divisible\$by\$2\$and\$3\$
7>\$Take\$the\$last\$digit,\$double\$it\$and\$subtract\$from\$the\$rest\$of\$the\$number.\$If\$the\$
For"example:5\$To\$check\$whether\$343\$is\$divisible\$by\$7.\$We\$double\$the\$last\$digit\$i.e\$3\$×\$
2\$=\$6\$and\$subtract\$it\$from\$the\$rest\$of\$the\$number\$i.e\$34.\$We\$get\$34\$–\$6\$=\$28.\$The\$
result\$28\$is\$divisible\$by\$7\$so\$is\$the\$original\$number\$i.e\$343\$is\$divisible\$by\$7\$
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Number'System'
The\$numbers\$can\$be\$defined\$in\$a\$lot\$of\$different\$ways\$like\$positive,\$negative,\$even,\$
odd,\$natural,\$whole,\$integers,\$fractions,\$etc.\$
This\$chapter\$deals\$with\$all\$these\$i.e\$different\$kinds\$of\$numbers.\$
\$
Positive/Negative'
Numbers\$can\$either\$be\$positive\$or\$negative\$or\$even\$none\$of\$the\$these!!\$
\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$
\$ >5\$ >4\$ >3\$ >2\$ >1\$ 0\$ 1\$ 2\$ 3\$ 4\$ 5\$
A\$number\$line\$illustrates\$this.\$
A\$negative\$number\$is\$defined\$as\$any\$number\$left\$to\$the\$zero\$or\$a\$number\$less\$than\$
zero.\$The\$symbol\$used\$to\$denote\$a\$negative\$number\$is\$-.\$
A\$positive\$number\$is\$defined\$as\$any\$number\$right\$to\$the\$zero\$or\$a\$number\$more\$than\$
zero.\$The\$symbol\$used\$to\$denote\$a\$positive\$number\$is\$+.\$
As\$we\$can\$see\$that\$the\$number\$zero\$is\$just\$used\$to\$make\$a\$distinction\$between\$
positive\$and\$negative\$number,\$so\$it\$is\$considered\$to\$be\$neither\$positive\$or\$negative\$
i.e\$zero'(0)'is'a'neutral'number.'
Things'to'keep'in'mind:0'
1. Positive\$×\$positive\$=\$positive\$
2. positive\$\$×\$negative\$=\$negative\$\$\$\$\$\$Multiplication\$
3. negative\$\$×\$negative\$=\$positive\$
4. positive/positive\$=\$positive\$\$\$\$\$\$\$\$\$\$\$
5. positive/negative\$=\$negative\$\$\$\$\$\$\$\$\$\$Division\$
6. negative/negative\$=\$positive\$
7. A\$double\$negative\$means\$positive.\$For\$example:>\$4\$–\$(>2)\$=\$4\$+\$2=\$6\$
A"few"more"definitions:>\$
1. Natural\$numbers:>\$1,\$2,\$3,\$4,\$5,\$………..(only\$positive)\$
2. Whole\$numbers:>\$0,\$1,\$2,\$3,\$………………(non\$–\$negative)\$
3. Integers:>……………….\$>5,\$>4,\$>3,\$>2,\$>1,\$0,\$1,\$2,\$3……………(negative/positive)\$
4. Fractions:>\$numbers\$of\$the\$form\$p/q\$where\$q\$?\$0\$
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Even/Odd'
Even:>\$An\$even\$number\$is\$an\$integer\$that\$is\$divisible\$by\$2.\$For\$example:>\$>24,\$>36,\$>20,\$
0,\$20,\$42,\$38\$etc.\$\$\$An\$even\$number\$can\$be\$written\$in\$the\$form\$n\$=\$2k,\$where\$k\$is\$an\$
integer.\$
Odd:>\$An\$odd\$number\$is\$an\$integer\$that\$is\$not\$divisible\$by\$2.\$For\$example:>\$>23,\$>37,\$>
19,\$1,\$3,\$17,\$etc.\$\$\$\$An\$odd\$number\$can\$be\$written\$in\$the\$form\$n\$=\$2k+1,\$where\$k\$is\$an\$
integer.\$
Things'to'remember:0\$
1. even\$+\$even\$=\$even\$
3. odd\$+\$odd\$=\$even\$
4. even\$–\$even\$=\$even\$
5. even\$–\$odd\$=\$odd\$\$\$\$\$\$\$\$\$\$\$\$Subtraction\$
6. odd\$–\$odd\$=\$even\$
7. even\$×\$even\$=\$even\$
8. even\$×\$odd\$=\$even\$\$\$\$\$\$\$\$\$Multiplication\$
9. odd\$×\$odd\$=\$odd\$
10. Division\$of\$even\$or\$odd\$numbers\$does\$not\$follow\$any\$specific\$rules.\$It\$may\$
result\$in\$an\$even\$or\$odd\$integer\$or\$a\$fraction.\$For\$example:\$>\$6/2\$=\$3,\$6/3\$=\$2,\$
35/5\$=\$7,\$6/4\$?\$integer\$
11. The\$only\$specific\$rule\$for\$division\$is\$(Odd/Even)\$?\$integer\$i.e\$an\$odd\$integer\$
when\$divided\$by\$an\$even\$integer\$would\$never\$result\$in\$an\$integer.\$
\$
Consecutive'Integers:A'
The\$word\$consecutive\$means\$one\$after\$the\$other.\$Similarly,\$consecutive\$numbers\$are\$
the\$numbers\$that\$follow\$one\$another\$from\$a\$given\$value.\$
For\$Example:>\$1,\$2,\$3,\$4\$are\$consecutive\$integers\$and\$\$\$\$\$\$\$\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$>12,\$>13,\$>14,\$>15,\$16\$are\$also\$consecutive\$integers.\$\$\$\$
Consecutive\$integers\$can\$also\$make\$some\$specific\$patterns\$like:>\$
• Consecutive\$even\$integers:>\$2,\$4,\$6,\$8,\$10………\$
• Consecutive\$odd\$integers:>\$1,\$3,\$5,\$7,\$9……\$
• Consecutive\$multiples\$of\$5:>\$5,\$10,\$15,\$20,\$25………..\$
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Things'to'remember:0'
1. The\$arithmetic\$mean\$(average)\$is\$equal\$to\$the\$median\$in\$a\$set\$of\$consecutive\$
numbers.\$
2. The\$average\$and\$the\$median\$are\$both\$equal\$to\$the\$average\$of\$the\$1
st
\$and\$last\$
numbers\$of\$the\$set.\$\$
For"Example:>\$In\$the\$set\$2,\$4,\$6,\$8……………..200,\$the\$average\$and\$the\$median\$are\$both\$
equal\$to\$the\$average\$of\$the\$1
st
\$and\$the\$last\$numbers\$i.e\$Average\$=\$median\$=\$
(2+200)/2=\$101\$
3. Two\$consecutive\$integers\$are\$never\$divisible\$by\$the\$same\$prime\$number\$and\$
therefore\$by\$the\$same\$number.\$\$\$
the\$result.\$
For\$example:>\$The\$number\$of\$integers\$from\$2\$to\$7\$is\$not\$5\$but\$6\$(2,\$3,\$,4,\$\$5,\$6\$
and\$7)\$
But\$the\$number\$of\$integers\$between\$2\$and\$7\$is\$5.\$
5. The\$product\$of\$n\$consecutive\$numbers\$is\$always\$divisible\$by\$n.\$
6. The\$sum\$of\$n\$consecutive\$integers\$is\$always\$divisible\$by\$n\$if\$n\$is\$odd\$and\$
never\$divisible\$by\$n\$if\$n\$is\$even.\$
For\$example:>\$The\$sum\$of\$1,\$2\$and\$3\$i.e\$6\$is\$divisible\$by\$3(number\$of\$integers\$
is\$odd)\$but\$he\$sum\$of\$1,\$2,\$3\$and\$4\$i.e\$10\$is\$not\$divisible\$by\$4(number\$of\$
integers\$is\$even)\$
Divisibility'Rules:A''
A\$few\$important\$divisibility\$rules:>\$
2\$>\$A\$number\$is\$divisible\$by\$2\$if\$the\$last\$digit\$of\$the\$number\$is\$even\$
3>\$A\$number\$is\$divisible\$by\$3\$if\$the\$sum\$of\$the\$digits\$of\$the\$number\$is\$divisible\$by\$3\$
4>\$A\$number\$is\$divisible\$by\$4\$if\$the\$last\$two\$digits\$form\$a\$number\$that\$is\$divisible\$by\$4\$
5>\$A\$number\$is\$divisible\$by\$5\$if\$the\$last\$digit\$is\$0\$or\$5\$
6>\$A\$number\$is\$divisible\$by\$6\$if\$the\$number\$is\$divisible\$by\$2\$and\$3\$
7>\$Take\$the\$last\$digit,\$double\$it\$and\$subtract\$from\$the\$rest\$of\$the\$number.\$If\$the\$
For"example:5\$To\$check\$whether\$343\$is\$divisible\$by\$7.\$We\$double\$the\$last\$digit\$i.e\$3\$×\$
2\$=\$6\$and\$subtract\$it\$from\$the\$rest\$of\$the\$number\$i.e\$34.\$We\$get\$34\$–\$6\$=\$28.\$The\$
result\$28\$is\$divisible\$by\$7\$so\$is\$the\$original\$number\$i.e\$343\$is\$divisible\$by\$7\$
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8>\$A\$number\$is\$divisible\$by\$8\$if\$the\$last\$three\$digits\$form\$a\$number\$that\$is\$divisible\$by\$
8\$
9>\$A\$number\$is\$divisible\$by\$9\$if\$the\$sum\$of\$the\$digits\$of\$the\$number\$is\$divisible\$by\$9\$
10\$>\$A\$number\$is\$divisible\$by\$10\$if\$the\$number\$ends\$in\$a\$0\$
12>\$A\$number\$is\$divisible\$by\$if\$the\$number\$is\$divisible\$by\$both\$3\$and\$4\$
25\$>\$A\$number\$is\$divisible\$by\$25\$if\$the\$number\$ends\$in\$00,\$25,\$50\$or\$75\$
\$
Prime'numbers:A'
A\$prime\$number\$is\$a\$positive\$integer\$which\$has\$exactly\$two\$factors\$i.e\$1\$and\$the\$
number\$itself.\$
Things'to'remember:0'
1. 1\$is\$not\$prime.\$
2. The\$smallest\$prime\$number\$is\$2\$
3. The\$only\$even\$prime\$number\$is\$2\$
4. All\$prime\$numbers\$except\$2\$and\$5\$end\$in\$1,\$3,\$7\$or\$9\$
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Number'System'
The\$numbers\$can\$be\$defined\$in\$a\$lot\$of\$different\$ways\$like\$positive,\$negative,\$even,\$
odd,\$natural,\$whole,\$integers,\$fractions,\$etc.\$
This\$chapter\$deals\$with\$all\$these\$i.e\$different\$kinds\$of\$numbers.\$
\$
Positive/Negative'
Numbers\$can\$either\$be\$positive\$or\$negative\$or\$even\$none\$of\$the\$these!!\$
\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$ |\$
\$ >5\$ >4\$ >3\$ >2\$ >1\$ 0\$ 1\$ 2\$ 3\$ 4\$ 5\$
A\$number\$line\$illustrates\$this.\$
A\$negative\$number\$is\$defined\$as\$any\$number\$left\$to\$the\$zero\$or\$a\$number\$less\$than\$
zero.\$The\$symbol\$used\$to\$denote\$a\$negative\$number\$is\$-.\$
A\$positive\$number\$is\$defined\$as\$any\$number\$right\$to\$the\$zero\$or\$a\$number\$more\$than\$
zero.\$The\$symbol\$used\$to\$denote\$a\$positive\$number\$is\$+.\$
As\$we\$can\$see\$that\$the\$number\$zero\$is\$just\$used\$to\$make\$a\$distinction\$between\$
positive\$and\$negative\$number,\$so\$it\$is\$considered\$to\$be\$neither\$positive\$or\$negative\$
i.e\$zero'(0)'is'a'neutral'number.'
Things'to'keep'in'mind:0'
1. Positive\$×\$positive\$=\$positive\$
2. positive\$\$×\$negative\$=\$negative\$\$\$\$\$\$Multiplication\$
3. negative\$\$×\$negative\$=\$positive\$
4. positive/positive\$=\$positive\$\$\$\$\$\$\$\$\$\$\$
5. positive/negative\$=\$negative\$\$\$\$\$\$\$\$\$\$Division\$
6. negative/negative\$=\$positive\$
7. A\$double\$negative\$means\$positive.\$For\$example:>\$4\$–\$(>2)\$=\$4\$+\$2=\$6\$
A"few"more"definitions:>\$
1. Natural\$numbers:>\$1,\$2,\$3,\$4,\$5,\$………..(only\$positive)\$
2. Whole\$numbers:>\$0,\$1,\$2,\$3,\$………………(non\$–\$negative)\$
3. Integers:>……………….\$>5,\$>4,\$>3,\$>2,\$>1,\$0,\$1,\$2,\$3……………(negative/positive)\$
4. Fractions:>\$numbers\$of\$the\$form\$p/q\$where\$q\$?\$0\$
'
PS  WC
WC  PD
Even/Odd'
Even:>\$An\$even\$number\$is\$an\$integer\$that\$is\$divisible\$by\$2.\$For\$example:>\$>24,\$>36,\$>20,\$
0,\$20,\$42,\$38\$etc.\$\$\$An\$even\$number\$can\$be\$written\$in\$the\$form\$n\$=\$2k,\$where\$k\$is\$an\$
integer.\$
Odd:>\$An\$odd\$number\$is\$an\$integer\$that\$is\$not\$divisible\$by\$2.\$For\$example:>\$>23,\$>37,\$>
19,\$1,\$3,\$17,\$etc.\$\$\$\$An\$odd\$number\$can\$be\$written\$in\$the\$form\$n\$=\$2k+1,\$where\$k\$is\$an\$
integer.\$
Things'to'remember:0\$
1. even\$+\$even\$=\$even\$
3. odd\$+\$odd\$=\$even\$
4. even\$–\$even\$=\$even\$
5. even\$–\$odd\$=\$odd\$\$\$\$\$\$\$\$\$\$\$\$Subtraction\$
6. odd\$–\$odd\$=\$even\$
7. even\$×\$even\$=\$even\$
8. even\$×\$odd\$=\$even\$\$\$\$\$\$\$\$\$Multiplication\$
9. odd\$×\$odd\$=\$odd\$
10. Division\$of\$even\$or\$odd\$numbers\$does\$not\$follow\$any\$specific\$rules.\$It\$may\$
result\$in\$an\$even\$or\$odd\$integer\$or\$a\$fraction.\$For\$example:\$>\$6/2\$=\$3,\$6/3\$=\$2,\$
35/5\$=\$7,\$6/4\$?\$integer\$
11. The\$only\$specific\$rule\$for\$division\$is\$(Odd/Even)\$?\$integer\$i.e\$an\$odd\$integer\$
when\$divided\$by\$an\$even\$integer\$would\$never\$result\$in\$an\$integer.\$
\$
Consecutive'Integers:A'
The\$word\$consecutive\$means\$one\$after\$the\$other.\$Similarly,\$consecutive\$numbers\$are\$
the\$numbers\$that\$follow\$one\$another\$from\$a\$given\$value.\$
For\$Example:>\$1,\$2,\$3,\$4\$are\$consecutive\$integers\$and\$\$\$\$\$\$\$\$
\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$>12,\$>13,\$>14,\$>15,\$16\$are\$also\$consecutive\$integers.\$\$\$\$
Consecutive\$integers\$can\$also\$make\$some\$specific\$patterns\$like:>\$
• Consecutive\$even\$integers:>\$2,\$4,\$6,\$8,\$10………\$
• Consecutive\$odd\$integers:>\$1,\$3,\$5,\$7,\$9……\$
• Consecutive\$multiples\$of\$5:>\$5,\$10,\$15,\$20,\$25………..\$
'
PS  WC
WC  PD
Things'to'remember:0'
1. The\$arithmetic\$mean\$(average)\$is\$equal\$to\$the\$median\$in\$a\$set\$of\$consecutive\$
numbers.\$
2. The\$average\$and\$the\$median\$are\$both\$equal\$to\$the\$average\$of\$the\$1
st
\$and\$last\$
numbers\$of\$the\$set.\$\$
For"Example:>\$In\$the\$set\$2,\$4,\$6,\$8……………..200,\$the\$average\$and\$the\$median\$are\$both\$
equal\$to\$the\$average\$of\$the\$1
st
\$and\$the\$last\$numbers\$i.e\$Average\$=\$median\$=\$
(2+200)/2=\$101\$
3. Two\$consecutive\$integers\$are\$never\$divisible\$by\$the\$same\$prime\$number\$and\$
therefore\$by\$the\$same\$number.\$\$\$
the\$result.\$
For\$example:>\$The\$number\$of\$integers\$from\$2\$to\$7\$is\$not\$5\$but\$6\$(2,\$3,\$,4,\$\$5,\$6\$
and\$7)\$
But\$the\$number\$of\$integers\$between\$2\$and\$7\$is\$5.\$
5. The\$product\$of\$n\$consecutive\$numbers\$is\$always\$divisible\$by\$n.\$
6. The\$sum\$of\$n\$consecutive\$integers\$is\$always\$divisible\$by\$n\$if\$n\$is\$odd\$and\$
never\$divisible\$by\$n\$if\$n\$is\$even.\$
For\$example:>\$The\$sum\$of\$1,\$2\$and\$3\$i.e\$6\$is\$divisible\$by\$3(number\$of\$integers\$
is\$odd)\$but\$he\$sum\$of\$1,\$2,\$3\$and\$4\$i.e\$10\$is\$not\$divisible\$by\$4(number\$of\$
integers\$is\$even)\$
Divisibility'Rules:A''
A\$few\$important\$divisibility\$rules:>\$
2\$>\$A\$number\$is\$divisible\$by\$2\$if\$the\$last\$digit\$of\$the\$number\$is\$even\$
3>\$A\$number\$is\$divisible\$by\$3\$if\$the\$sum\$of\$the\$digits\$of\$the\$number\$is\$divisible\$by\$3\$
4>\$A\$number\$is\$divisible\$by\$4\$if\$the\$last\$two\$digits\$form\$a\$number\$that\$is\$divisible\$by\$4\$
5>\$A\$number\$is\$divisible\$by\$5\$if\$the\$last\$digit\$is\$0\$or\$5\$
6>\$A\$number\$is\$divisible\$by\$6\$if\$the\$number\$is\$divisible\$by\$2\$and\$3\$
7>\$Take\$the\$last\$digit,\$double\$it\$and\$subtract\$from\$the\$rest\$of\$the\$number.\$If\$the\$
For"example:5\$To\$check\$whether\$343\$is\$divisible\$by\$7.\$We\$double\$the\$last\$digit\$i.e\$3\$×\$
2\$=\$6\$and\$subtract\$it\$from\$the\$rest\$of\$the\$number\$i.e\$34.\$We\$get\$34\$–\$6\$=\$28.\$The\$
result\$28\$is\$divisible\$by\$7\$so\$is\$the\$original\$number\$i.e\$343\$is\$divisible\$by\$7\$
PS  WC
WC  PD
8>\$A\$number\$is\$divisible\$by\$8\$if\$the\$last\$three\$digits\$form\$a\$number\$that\$is\$divisible\$by\$
8\$
9>\$A\$number\$is\$divisible\$by\$9\$if\$the\$sum\$of\$the\$digits\$of\$the\$number\$is\$divisible\$by\$9\$
10\$>\$A\$number\$is\$divisible\$by\$10\$if\$the\$number\$ends\$in\$a\$0\$
12>\$A\$number\$is\$divisible\$by\$if\$the\$number\$is\$divisible\$by\$both\$3\$and\$4\$
25\$>\$A\$number\$is\$divisible\$by\$25\$if\$the\$number\$ends\$in\$00,\$25,\$50\$or\$75\$
\$
Prime'numbers:A'
A\$prime\$number\$is\$a\$positive\$integer\$which\$has\$exactly\$two\$factors\$i.e\$1\$and\$the\$
number\$itself.\$
Things'to'remember:0'
1. 1\$is\$not\$prime.\$
2. The\$smallest\$prime\$number\$is\$2\$
3. The\$only\$even\$prime\$number\$is\$2\$
4. All\$prime\$numbers\$except\$2\$and\$5\$end\$in\$1,\$3,\$7\$or\$9\$
'
'
'
'
'
'
'
'
'
'
PS  WC
WC  PD
Factors/Multiples:A'
A\$factor\$is\$a\$positive\$integer\$that\$divides\$evenly\$into\$an\$integer.\$In\$general,\$it\$is\$said\$
‘m’\$is\$a\$factor\$of\$‘n’,\$for\$non>zero\$integers\$m\$and\$n,\$if\$there\$exists\$a\$relation\$such\$that\$
n/m\$=\$k,\$where\$k\$is\$an\$integer.\$
A\$multiple\$is\$an\$integer\$that\$can\$be\$evenly\$divided\$into\$an\$integer.\$In\$general,\$it\$is\$
said\$that\$‘m’\$is\$a\$multiple\$of\$‘n’,\$for\$non>zero\$integers\$m\$and\$n,\$if\$there\$exists\$a\$
relation\$such\$that\$m\$=\$nk,\$where\$k\$is\$an\$integer.\$
Things'to'remember:0'
1. 1\$is\$a\$factor\$of\$every\$integer\$
2. Every\$integer\$is\$a\$factor\$and\$a\$multiple\$of\$itself.\$It\$is\$the\$smallest\$positive\$
multiple\$of\$itself\$and\$the\$largest\$positive\$factor\$of\$itself.\$
3. If\$‘x’\$is\$a\$factor\$of\$‘y’\$and\$‘y’\$is\$a\$factor\$of\$‘z’,\$then\$‘x’\$is\$a\$factor\$of\$‘z’\$
4. If\$‘x’\$is\$a\$factor\$of\$‘y’\$and\$‘x’\$is\$a\$factor\$of\$‘z’,\$then\$‘x’\$would\$be\$a\$factor\$of\$(y+z)\$
5. If\$‘x’\$is\$a\$factor\$of\$‘y’\$and\$‘y’\$is\$a\$factor\$of\$x,\$then\$x\$=\$y\$or\$x\$=\$>y\$
6. All\$numbers\$have\$a\$limited\$number\$of\$factors\$and\$an\$unlimited\$number\$of\$
multiples\$
\$
Number'of'factors:A'
As\$we\$discussed,\$all\$the\$numbers\$have\$a\$limited\$number\$of\$factors\$so\$we\$can\$be\$
mentioned\$steps:>\$
1) Make\$the\$prime\$factorization\$of\$the\$integer\$i.e\$write\$the\$integer\$in\$the\$form\$\$
n\$=\$a
p
\$×\$b
q
\$×\$c
r
……..\$,\$where\$a,\$b,\$c\$are\$the\$prime\$factors\$of\$n\$and\$p,\$q,\$r\$are\$their\$
powers.\$
2) The\$number\$of\$factors\$of\$n\$will\$be\$given\$by\$(p+1)(q+1)(r+1)……….\$
Example:5\$What\$is\$the\$number\$of\$factors\$of\$441?\$
Sol:5\$15435\$=\$3
2
\$×\$5
1
\$×\$7
3\$\$\$\$\$
So,\$the\$number\$of\$factors\$will\$be\$(2+1)(1+1)(3+1)\$=\$3\$×\$2\$×\$4\$=\$24\$
\$
```
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