Number System Notes | EduRev

Created by: Wizius Careers

: Number System Notes | EduRev

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PS  WC 
WC  PD 
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$$$$$$$$$$$
5. positive/negative$=$negative$$$$$$$$$$Division$
6. negative/negative$=$positive$
7. A$double$negative$means$positive.$For$example:>$4$–$(>2)$=$4$+$2=$6$
A"few"more"definitions:>$
1. Natural$numbers:>$1,$2,$3,$4,$5,$………..(only$positive)$
2. Whole$numbers:>$0,$1,$2,$3,$………………(non$–$negative)$
3. Integers:>……………….$>5,$>4,$>3,$>2,$>1,$0,$1,$2,$3……………(negative/positive)$
4. Fractions:>$numbers$of$the$form$p/q$where$q$?$0$
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PS  WC 
WC  PD 
Number'System'
The$numbers$can$be$defined$in$a$lot$of$different$ways$like$positive,$negative,$even,$
odd,$natural,$whole,$integers,$fractions,$etc.$
This$chapter$deals$with$all$these$i.e$different$kinds$of$numbers.$
$
Positive/Negative'
Numbers$can$either$be$positive$or$negative$or$even$none$of$the$these!!$
$ |$ |$ |$ |$ |$ |$ |$ |$ |$ |$ |$
$ >5$ >4$ >3$ >2$ >1$ 0$ 1$ 2$ 3$ 4$ 5$
A$number$line$illustrates$this.$
A$negative$number$is$defined$as$any$number$left$to$the$zero$or$a$number$less$than$
zero.$The$symbol$used$to$denote$a$negative$number$is$-.$
A$positive$number$is$defined$as$any$number$right$to$the$zero$or$a$number$more$than$
zero.$The$symbol$used$to$denote$a$positive$number$is$+.$
As$we$can$see$that$the$number$zero$is$just$used$to$make$a$distinction$between$
positive$and$negative$number,$so$it$is$considered$to$be$neither$positive$or$negative$
i.e$zero'(0)'is'a'neutral'number.'
Things'to'keep'in'mind:0'
1. Positive$×$positive$=$positive$
2. positive$$×$negative$=$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.$$$$An$odd$number$can$be$written$in$the$form$n$=$2k+1,$where$k$is$an$
integer.$
Things'to'remember:0$
1. even$+$even$=$even$
2. even$+$odd$=$odd$$$$$$$$$$$$Addition$
3. odd$+$odd$=$even$
4. even$–$even$=$even$
5. even$–$odd$=$odd$$$$$$$$$$$$Subtraction$
6. odd$–$odd$=$even$
7. even$×$even$=$even$
8. even$×$odd$=$even$$$$$$$$$Multiplication$
9. odd$×$odd$=$odd$
10. Division$of$even$or$odd$numbers$does$not$follow$any$specific$rules.$It$may$
result$in$an$even$or$odd$integer$or$a$fraction.$For$example:$>$6/2$=$3,$6/3$=$2,$
35/5$=$7,$6/4$?$integer$
11. The$only$specific$rule$for$division$is$(Odd/Even)$?$integer$i.e$an$odd$integer$
when$divided$by$an$even$integer$would$never$result$in$an$integer.$
$
Consecutive'Integers:A'
The$word$consecutive$means$one$after$the$other.$Similarly,$consecutive$numbers$are$
the$numbers$that$follow$one$another$from$a$given$value.$
For$Example:>$1,$2,$3,$4$are$consecutive$integers$and$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$>12,$>13,$>14,$>15,$16$are$also$consecutive$integers.$$$$
Consecutive$integers$can$also$make$some$specific$patterns$like:>$
• Consecutive$even$integers:>$2,$4,$6,$8,$10………$
• Consecutive$odd$integers:>$1,$3,$5,$7,$9……$
• Consecutive$multiples$of$5:>$5,$10,$15,$20,$25………..$
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PS  WC 
WC  PD 
Number'System'
The$numbers$can$be$defined$in$a$lot$of$different$ways$like$positive,$negative,$even,$
odd,$natural,$whole,$integers,$fractions,$etc.$
This$chapter$deals$with$all$these$i.e$different$kinds$of$numbers.$
$
Positive/Negative'
Numbers$can$either$be$positive$or$negative$or$even$none$of$the$these!!$
$ |$ |$ |$ |$ |$ |$ |$ |$ |$ |$ |$
$ >5$ >4$ >3$ >2$ >1$ 0$ 1$ 2$ 3$ 4$ 5$
A$number$line$illustrates$this.$
A$negative$number$is$defined$as$any$number$left$to$the$zero$or$a$number$less$than$
zero.$The$symbol$used$to$denote$a$negative$number$is$-.$
A$positive$number$is$defined$as$any$number$right$to$the$zero$or$a$number$more$than$
zero.$The$symbol$used$to$denote$a$positive$number$is$+.$
As$we$can$see$that$the$number$zero$is$just$used$to$make$a$distinction$between$
positive$and$negative$number,$so$it$is$considered$to$be$neither$positive$or$negative$
i.e$zero'(0)'is'a'neutral'number.'
Things'to'keep'in'mind:0'
1. Positive$×$positive$=$positive$
2. positive$$×$negative$=$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.$$$$An$odd$number$can$be$written$in$the$form$n$=$2k+1,$where$k$is$an$
integer.$
Things'to'remember:0$
1. even$+$even$=$even$
2. even$+$odd$=$odd$$$$$$$$$$$$Addition$
3. odd$+$odd$=$even$
4. even$–$even$=$even$
5. even$–$odd$=$odd$$$$$$$$$$$$Subtraction$
6. odd$–$odd$=$even$
7. even$×$even$=$even$
8. even$×$odd$=$even$$$$$$$$$Multiplication$
9. odd$×$odd$=$odd$
10. Division$of$even$or$odd$numbers$does$not$follow$any$specific$rules.$It$may$
result$in$an$even$or$odd$integer$or$a$fraction.$For$example:$>$6/2$=$3,$6/3$=$2,$
35/5$=$7,$6/4$?$integer$
11. The$only$specific$rule$for$division$is$(Odd/Even)$?$integer$i.e$an$odd$integer$
when$divided$by$an$even$integer$would$never$result$in$an$integer.$
$
Consecutive'Integers:A'
The$word$consecutive$means$one$after$the$other.$Similarly,$consecutive$numbers$are$
the$numbers$that$follow$one$another$from$a$given$value.$
For$Example:>$1,$2,$3,$4$are$consecutive$integers$and$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$>12,$>13,$>14,$>15,$16$are$also$consecutive$integers.$$$$
Consecutive$integers$can$also$make$some$specific$patterns$like:>$
• Consecutive$even$integers:>$2,$4,$6,$8,$10………$
• Consecutive$odd$integers:>$1,$3,$5,$7,$9……$
• Consecutive$multiples$of$5:>$5,$10,$15,$20,$25………..$
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Things'to'remember:0'
1. The$arithmetic$mean$(average)$is$equal$to$the$median$in$a$set$of$consecutive$
numbers.$
2. The$average$and$the$median$are$both$equal$to$the$average$of$the$1
st
$and$last$
numbers$of$the$set.$$
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.$$$
4. To$count$the$number$of$integers$from$a$to$b,$subtract$a$from$b$and$add$1$to$
the$result.$
For$example:>$The$number$of$integers$from$2$to$7$is$not$5$but$6$(2,$3,$,4,$$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$
answer$is$divisible$by$7,$then$the$number$is$divisible$by$7.$
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.$$$$An$odd$number$can$be$written$in$the$form$n$=$2k+1,$where$k$is$an$
integer.$
Things'to'remember:0$
1. even$+$even$=$even$
2. even$+$odd$=$odd$$$$$$$$$$$$Addition$
3. odd$+$odd$=$even$
4. even$–$even$=$even$
5. even$–$odd$=$odd$$$$$$$$$$$$Subtraction$
6. odd$–$odd$=$even$
7. even$×$even$=$even$
8. even$×$odd$=$even$$$$$$$$$Multiplication$
9. odd$×$odd$=$odd$
10. Division$of$even$or$odd$numbers$does$not$follow$any$specific$rules.$It$may$
result$in$an$even$or$odd$integer$or$a$fraction.$For$example:$>$6/2$=$3,$6/3$=$2,$
35/5$=$7,$6/4$?$integer$
11. The$only$specific$rule$for$division$is$(Odd/Even)$?$integer$i.e$an$odd$integer$
when$divided$by$an$even$integer$would$never$result$in$an$integer.$
$
Consecutive'Integers:A'
The$word$consecutive$means$one$after$the$other.$Similarly,$consecutive$numbers$are$
the$numbers$that$follow$one$another$from$a$given$value.$
For$Example:>$1,$2,$3,$4$are$consecutive$integers$and$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$>12,$>13,$>14,$>15,$16$are$also$consecutive$integers.$$$$
Consecutive$integers$can$also$make$some$specific$patterns$like:>$
• Consecutive$even$integers:>$2,$4,$6,$8,$10………$
• Consecutive$odd$integers:>$1,$3,$5,$7,$9……$
• Consecutive$multiples$of$5:>$5,$10,$15,$20,$25………..$
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Things'to'remember:0'
1. The$arithmetic$mean$(average)$is$equal$to$the$median$in$a$set$of$consecutive$
numbers.$
2. The$average$and$the$median$are$both$equal$to$the$average$of$the$1
st
$and$last$
numbers$of$the$set.$$
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.$$$
4. To$count$the$number$of$integers$from$a$to$b,$subtract$a$from$b$and$add$1$to$
the$result.$
For$example:>$The$number$of$integers$from$2$to$7$is$not$5$but$6$(2,$3,$,4,$$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$
answer$is$divisible$by$7,$then$the$number$is$divisible$by$7.$
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$$$$$$$$$$$
5. positive/negative$=$negative$$$$$$$$$$Division$
6. negative/negative$=$positive$
7. A$double$negative$means$positive.$For$example:>$4$–$(>2)$=$4$+$2=$6$
A"few"more"definitions:>$
1. Natural$numbers:>$1,$2,$3,$4,$5,$………..(only$positive)$
2. Whole$numbers:>$0,$1,$2,$3,$………………(non$–$negative)$
3. Integers:>……………….$>5,$>4,$>3,$>2,$>1,$0,$1,$2,$3……………(negative/positive)$
4. Fractions:>$numbers$of$the$form$p/q$where$q$?$0$
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Even/Odd'
Even:>$An$even$number$is$an$integer$that$is$divisible$by$2.$For$example:>$>24,$>36,$>20,$
0,$20,$42,$38$etc.$$$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.$$$$An$odd$number$can$be$written$in$the$form$n$=$2k+1,$where$k$is$an$
integer.$
Things'to'remember:0$
1. even$+$even$=$even$
2. even$+$odd$=$odd$$$$$$$$$$$$Addition$
3. odd$+$odd$=$even$
4. even$–$even$=$even$
5. even$–$odd$=$odd$$$$$$$$$$$$Subtraction$
6. odd$–$odd$=$even$
7. even$×$even$=$even$
8. even$×$odd$=$even$$$$$$$$$Multiplication$
9. odd$×$odd$=$odd$
10. Division$of$even$or$odd$numbers$does$not$follow$any$specific$rules.$It$may$
result$in$an$even$or$odd$integer$or$a$fraction.$For$example:$>$6/2$=$3,$6/3$=$2,$
35/5$=$7,$6/4$?$integer$
11. The$only$specific$rule$for$division$is$(Odd/Even)$?$integer$i.e$an$odd$integer$
when$divided$by$an$even$integer$would$never$result$in$an$integer.$
$
Consecutive'Integers:A'
The$word$consecutive$means$one$after$the$other.$Similarly,$consecutive$numbers$are$
the$numbers$that$follow$one$another$from$a$given$value.$
For$Example:>$1,$2,$3,$4$are$consecutive$integers$and$$$$$$$$
$$$$$$$$$$$$$$$$$$$$$$$$$$>12,$>13,$>14,$>15,$16$are$also$consecutive$integers.$$$$
Consecutive$integers$can$also$make$some$specific$patterns$like:>$
• Consecutive$even$integers:>$2,$4,$6,$8,$10………$
• Consecutive$odd$integers:>$1,$3,$5,$7,$9……$
• Consecutive$multiples$of$5:>$5,$10,$15,$20,$25………..$
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Things'to'remember:0'
1. The$arithmetic$mean$(average)$is$equal$to$the$median$in$a$set$of$consecutive$
numbers.$
2. The$average$and$the$median$are$both$equal$to$the$average$of$the$1
st
$and$last$
numbers$of$the$set.$$
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.$$$
4. To$count$the$number$of$integers$from$a$to$b,$subtract$a$from$b$and$add$1$to$
the$result.$
For$example:>$The$number$of$integers$from$2$to$7$is$not$5$but$6$(2,$3,$,4,$$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$
answer$is$divisible$by$7,$then$the$number$is$divisible$by$7.$
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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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$be$
asked$to$find$the$numbers$of$factors$of$a$number.$
If$you$are$asked$to$find$the$number$of$factors$of$an$integer,$follow$the$below$
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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