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**Exercise Page: 5**

**In questions 1 to 38, out of the four options, only one is correct. Write the correct answer.****Q.1. The product of the place values of two 2’s in 428721 is****(a) 4 ****(b) 40000 ****(c) 400000 ****(d) 40000000****Ans: **(c)**Explanation****:**

The product of the place values of two 2’s in 428721 is

There are two 2’s in the given number.

So, the first 2 is in the tenth place,

Then, the product is = 2 × 10

= 20

The other 2 is in place value of ten thousand.

Then, = 2 × 10000

= 20000

Therefore, the product of place values = 20 × 20000

= 400000**Q.2. 3 × 10000 + 7 × 1000 + 9 × 100 + 0 ×10 + 4 is the same as****(a) 3794 ****(b) 37940 ****(c) 37904****(d) 379409****Ans: **(c)**Explanation****:**

3 × 10000 = 30000

The place value is 30000

7 × 1000 = 7000

The place value is 7000

9 × 100 = 900

The place value is 900

0 × 10 = 0

4 = 4

Therefore, the sum of all place value is = 30000 + 7000 + 900 + 0 + 4

= 37904**Q.3. If 1 is added to the greatest 7- digit number, it will be equal to****(a) 10 thousand ****(b) 1 lakh ****(c) 10 lakh ****(d) 1 crore****Ans: **(d)**Explanation****:**

We know that, the greatest number is 99,99,999

Then, 1 is added to 99,99,999 = 99,99,999 + 1

= 1,00,00,000

= 1 crore**Q.4. The expanded form of the number 9578 is****(a) 9 × 10000 + 5 × 1000 + 7 × 10 + 8 × 1****(b) 9 × 1000 + 5 × 100 + 7 × 10 + 8 × 1****(c) 9 × 1000 + 57 × 10 + 8 × 1****(d) 9 × 100 + 5 × 100 + 7 × 10 + 8 × 1****Ans: **(b)**Explanation****:**

Consider the given number 9578,

The place value of 8 is ones = 8 × 1

The place value of 7 is tens = 7 × 10

The place value of 5 is thousand = 5 × 100

The place value of 9 is ten thousand = 9 × 1000**Q.5. When rounded off to nearest thousands, the number 85642 is****(a) 85600 ****(b) 85700 ****(c) 85000 ****(d) 86000****Ans: **(d)**Explanation****:**** **When rounded off to nearest thousands, the number 85642 is = 86000**Q.6. The largest 4-digit number, using any one digit twice, from digits 5, 9, 2 and 6 is****(a) 9652 ****(b) 9562 ****(c) 9659 ****(d) 9965****Ans: **(d)**Explanation****:**** **Using 9 as twice from 5, 9, 2 and 6, then then number is 9965.**Q.7. In Indian System of Numeration, the number 58695376 is written as****(a) 58,69, 53, 76 ****(b) 58,695,376****(c) 5,86,95,376 ****(d) 586,95,376****Ans: **(c)**Explanation****:**** **In Indian System of Numeration, the number 58695376 is written as 5 crore, eighty six lakh, ninety five thousand, three hundred and seventy six = 5,86,95,376**Q.8. One million is equal to****(a) 1 lakh ****(b) 10 lakh ****(c) 1 crore ****(d) 10 crore****Ans: **(b)**Explanation****:**

One million is equal to ten lakh.

1,000,000 = 10,00,000**Q.9. The greatest number which on rounding off to nearest thousands gives 5000, is****(a) 5001 ****(b) 5559 ****(c) 5999 ****(d) 5499****Ans: **(d)**Explanation****:**** **The greatest number which on rounding off to nearest thousands gives 5000, is 5499.**Q.10. Keeping the place of 6 in the number 6350947 same, the smallest number obtained by rearranging other digits is****(a) 6975430 ****(b) 6043579 ****(c) 6034579 ****(d) 6034759****Ans: **(c)**Explanation****:**** **Keeping the place of 6 in the number 6350947 same, the smallest number obtained by rearranging other digits is 6034579.**Q.11. Which of the following numbers in Roman numerals is incorrect?****(a) LXXX ****(b) LXX ****(c) LX ****(d) LLX****Ans: **(d)**Explanation****:**** **As we know that, the symbol L can never be repeated.

Therefore, LLX is incorrect.**Q.12. The largest 5-digit number having three different digits is****(a) 98978 ****(b) 99897 ****(c) 99987 ****(d) 98799****Ans: **(c)**Explanation****:**** **In the given, options there are three numbers used 9, 8 and 7

To get the largest of 5- digit we have to arrange the numbers in descending order.

Then, from the given options 99987 is the largest of 5 – digit number.**Q.13. The smallest 4-digit number having three different digits is****(a) 1102 ****(b) 1012 ****(c) 1020 ****(d) 1002****Ans:** (d)**Explanation****:**** **In the given, options there are three numbers used 0, 1 and 2

To get the largest of 4- digit we have to arrange the numbers in ascending order.

Then, from the given options 1002 is the smallest of 4 – digit number.**Q.14. Number of whole numbers between 38 and 68 is****(a) 31 ****(b) 30 ****(c) 29 ****(d) 28****Ans: **(c)**Explanation****:** Number of whole numbers between 38 and 68 is 29.**Q.15. The product of successor and predecessor of 999 is****(a) 999000 ****(b) 998000 ****(c) 989000****(d) 1998****Ans: **(b)**Explanation****: **The number which comes immediately before a particular number is called its predecessor.

The successor of a whole number is the number obtained by adding 1 to it.

So, Successor of 999 = 999 + 1 = 1000

Predecessor = 999 – 1 = 998

Then, product of successor and predecessor of 999 is = 1000 × 998

= 998000**Q.16. The product of a non-zero whole number and its successor is always****(a) an even number ****(b) an odd number****(c) a prime number ****(d) divisible by 3****Ans: **(a)**Explanation****: **The product of a non-zero whole number and its successor is always an even number.

For example: – 4 × 5 = 20, 7 × 8 = 56**Q.17. A whole number is added to 25 and the same number is subtracted from 25. The sum of the resulting numbers is****(a) 0 ****(b) 25 ****(c) 50 ****(d) 75****Ans: **(c)**Explanation****: **

Let us assume the number be x.

From the question it is given that, number is added to 25 = x + 25

The same number is subtracted to from 25 = 25 – x

Then, the sum of the resulting numbers is = (x + 25) + (25 – x)

= x + 25 + 25 – x

= 50 + x – x

= 50 + 0

= 50**Q.18. Which of the following is not true?****(a) (7 + 8) + 9 = 7 + (8 + 9)****(b) (7 × 8) × 9 = 7 × (8 × 9)****(c) 7 + 8 × 9 = (7 + 8) × (7 + 9)****(d) 7 × (8 + 9) = (7 × 8) + (7 × 9)****Ans: **(c)**Explanation****:**

Consider the left hand side = 7 + 8 × 9

= 7 + (8 × 9)

= 7 + 72

= 79

Now, consider the right hand side = (7 + 8) × (7 + 9)

= 15 × 16

= 240

By comparing LHS and RHS

LHS ≠ RHS

79 ≠ 240**Q.19. By using dot (.) patterns, which of the following numbers can be arranged in all the three ways namely a line, a triangle and a rectangle?****(a) 9 ****(b) 10 ****(c) 11 ****(d) 12****Ans: **(b)**Explanation****:****Q.20. Which of the following statements is not true?****(a) Both addition and multiplication are associative for whole numbers.****(b) Zero is the identity for multiplication of whole numbers.****(c) Addition and multiplication both are commutative for whole numbers.****(d) Multiplication is distributive over addition for whole numbers.****Ans: **(b)**Q.21. Which of the following statements is not true?****(a) 0 + 0 = 0 ****(b) 0 – 0 = 0****(c) 0 × 0 = 0 ****(d) 0 ÷ 0 = 0****Ans:** (d)**Explanation****:**** **Zero divided by zero is not defined.**Q.22. The predecessor of 1 lakh is****(a) 99000 ****(b) 99999 ****(c) 999999 ****(d) 100001****Ans: **(b)**Explanation****:**

The number which comes immediately before a particular number is called its predecessor.

The predecessor of 1 lakh is = 1,00,000 – 1

= 99,999**Q.23. The successor of 1 million is****(a) 2 millions ****(b) 1000001 ****(c) 100001 ****(d) 10001****Ans: **(b)**Explanation****:**

The successor of a whole number is the number obtained by adding 1 to it.

We know that, 1 million = 10,00,000

Then, successor = 10,00,000 + 1

= 10,00,001**Q.24. Number of even numbers between 58 and 80 is****(a) 10 ****(b) 11 ****(c) 12 ****(d) 13****Ans: **(a)**Explanation****:** Even numbers between 58 and 80 are 60, 62, 64, 66, 68, 70, 72, 74, 76, 78.**Q.25. Sum of the number of primes between 16 to 80 and 90 to 100 is****(a) 20 ****(b) 18 ****(c) 17 ****(d) 16****Ans:** (c)**Explanation****: **

Prime numbers between 16 to 80 = 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73 and 79.

Then, number of primes between 16 to 80 = 16

Prime numbers between 90 to 100 = 97

Then, number of primes between 90 to 100 = 1

Therefore, Sum of the number of primes between 16 to 80 and 90 to 100 is,

= 16 + 1

= 17**Q.26. Which of the following statements is not true?****(a) The HCF of two distinct prime numbers is 1****(b) The HCF of two co prime numbers is 1****(c) The HCF of two consecutive even numbers is 2****(d) The HCF of an even and an odd number is even.****Ans: **(d)**Explanation****:** The HCF of an even and an odd number is odd number.**Q.27. The number of distinct prime factors of the largest 4-digit number is****(a) 2 ****(b) 3 ****(c) 5 ****(d) 11****Ans: **(b)**Explanation****:**** **

The largest 4 – digit number = 9999

Prime factors of 9999 = 3 × 3 × 11 × 101

So, 9999 = 3^{2} × 11 × 101

Therefore, distinct prime factors are = 3, 11 and 101

Number of distinct prime factors of the largest 4-digit number is = 3**Q.28. The number of distinct prime factors of the smallest 5-digit number is****(a) 2 ****(b) 4 ****(c) 6****(d) 8****Ans: **(a)**Explanation****:**

The smallest 5 – digit number = 10000

Prime factors of 10000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5

So, 10000 = 2^{4} × 5^{4}

Therefore, distinct prime factors are = 2 and 5

Number of distinct prime factors of the smallest 5-digit number is = 2**Q.29. If the number 7254*98 is divisible by 22, the digit at * is****(a) 1 ****(b) 2 ****(c) 6 ****(d) 0****Ans: **(c)**Explanation****:**

Take the alternating sum of the digits in the number, read from left to right. If that is divisible by 11, then the original number.

7 – 2 + 5 – 4 + * – 9 + 8 = (5 + *)

For the given number 7254 * 98 to be divisible by 11,

(5 + *) must also be divisible by 11

So, 5 + * = 11

Therefore * = 11 – 5

= 6**Q.30. The largest number which always divides the sum of any pair of consecutive odd numbers is****(a) 2 ****(b) 4 ****(c) 6 ****(d) 8****Ans: **(b)**Explanation****:**

1 + 3 = 4 = 4/4 = 1

3 + 5 = 8 = 8/4 = 2

The largest number which always divides the sum of any pair of consecutive odd numbers is 4.**Q.31. A number is divisible by 5 and 6. It may not be divisible by****(a) 10 ****(b) 15 ****(c) 30 ****(d) 60****Ans: **(d)**Explanation****:**

The LCM of 6 and 5 is 30.

So, 30 is divisible by 10, 15 and 30 in the given options.

But, 30 is not divisible by 60.**Q.32. The sum of the prime factors of 1729 is****(a) 13 ****(b) 19 ****(c) 32 ****(d) 39****Ans: **(d)**Explanation****:**

The prime factors of 1729 = 7 × 13 × 19

Therefore, the sum of prime numbers = 7 + 13 + 19

= 39**Q.33. The greatest number which always divides the product of the predecessor and successor of an odd natural number other than 1, is****(a) 6 ****(b) 4 ****(c) 16 ****(d) 8****Ans: **(b)**Explanation****:**

Let us assume an odd natural number be 5.

Then, predecessor of 5 = 5 -1 = 4

Successor of 5 = 5 + 1 = 6

Then, the product of predecessor and successor = 4 × 6

= 24

24 is divided by 4 = 24/4 = 6

Therefore, the greatest number which always divides the product of the predecessor and successor of an odd natural number other than 1, is 4.**Q.34. The number of common prime factors of 75, 60, 105 is****(a) 2 ****(b) 3 ****(c) 4 ****(d) 5****Ans:** (a)**Explanation****:**

Prime factors of,

75 = 3 × 5 × 5

60 = 2 × 2 × 3 × 5

105 = 3 × 5 × 7

So, common prime factors in the given three numbers are 3 and 5.

Therefore, the number of common prime factors of 75, 60, 105 is 2.**Q.35. Which of the following pairs is not coprime?****(a) 8, 10 ****(b) 11, 12 ****(c) 1, 3 ****(d) 31, 33****Ans: **(a)**Explanation****:**

First of all, both the numbers are even.

Then, common factor of both numbers is 2 other than 1.

Therefore, 8 and 10 are not coprime.**Q.36. Which of the following numbers is divisible by 11?****(a) 1011011 ****(b) 1111111 ****(c) 22222222 ****(d) 3333333****Ans: **(c)**Explanation****:**

To check the divisibility of a number by 11, the rule is, find the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) of the number. If the difference is either 0 or divisible by 11, then the number is divisible by 11.

So, 2 – 2 + 2 – 2 + 2 – 2 + 2 – 2 = 0

Therefore, 22222222 is divisible by 11.**Q.****37. LCM of 10, 15 and 20 is****(a) 30 (b) 60 (c) 90 (d) 180 Ans:** (b)

Factors of 10, 15 and 30 is,

Then, LCM of 10, 15 and 30 is 2 × 2 × 3 × 5 × 1 = 60

(a) 45

(b) 60

(c) 75

(d) 90

Ans:

Factors of 180.

Then, factors of 180 = 2 × 2 × 3 × 3 × 5

180 is not divided by 180.

Therefore, 75 is not the HCF of the number 180.

As per the rule of the Roman numerals, a symbol is not repeated more than three times.

If a symbol is repeated, its value is added as many times as it occurs: i.e. II is equal 2, XX is 20 and XXX is 30.

Left Hand Side = 5555

Right Hand Side = 5 × 1000 + 5 × 100 + 5 × 10 + 5 × 1

= 5000 + 500 + 50 + 5

= 5555

Left Hand Side = Right Hand Side

Left Hand Side = 39746

Right Hand Side = 3 × 10000 + 9 × 1000 + 7 × 100 + 4 × 10 + 6

= 30000 + 9000 + 700 + 40 + 6

= 39746

Left Hand Side = Right Hand Side

Left Hand Side = 82546

Right Hand Side = 8 × 1000 + 2 × 1000 + 5 × 100 + 4 × 10 + 6

= 8000 + 2000 + 500 + 40 + 6

= 10,546

Left Hand Side ≠ Right Hand Side

Left Hand Side = 532235

Right Hand Side = 5 × 100000 + 3 × 10000 + 2 × 1000 + 2 × 100 + 3 × 10 + 5

= 5,00,000 + 30,000 + 2000 + 200 + 30 + 5

= 5,32,235

Left Hand Side = Right Hand Side

Where, X = 10

IX = 9

So, XXIX = 10 + 10 + 9

= 29

Where, L = 50

X = 10

IV = 4

So, LXXIV = 50 + 10 + 10 + 4

= 74

Where, L = 50

IV = 4

VI = 6

So, LIV = 50 + 4 = 54

LVI = 50 + 6

Therefore, 54 < 56

Hence, LIV < LVI

In the question, the arrangement of the numbers in ascending order.

Descending order of the given number = 5487, 5478, 4587, 4578.

The number 85764 rounded off to nearest hundreds is written as 85800.

The number 7826 rounded off to nearest hundreds is written as 7800.

The number 12469 rounded off to nearest hundreds is written as 12500

So, sum of numbers after rounded off to hundreds = 7800 + 12500 = 20,300

Therefore, 20,300 is nearest to 20,000.

The largest six digit telephone number that can be formed by using digits 5, 3, 4, 7, 0, 8 only once is 875430.

The given number 8,16,52,318 will be read as eight crore sixteen lakh fifty two thousand three hundred and eighteen.

Among kilo, milli and centi, the smallest is milli.

The successor of a whole number is the number obtained by adding 1 to it.

Example: – consider the number 9 it is a one digit, then its successor = 9 + 1 = 10.

The successor of a whole number is the number obtained by adding 1 to it.

Example: – consider 3-digit number 999 it is a one digit, then its successor = 999 + 1 = 1000.

The number which comes immediately before a particular number is called its predecessor.

Example: – consider 2-digit number 10, then its predecessor = 10 – 1 = 9.

Consider the whole number 0,

Then, its predecessor = 0 – 1 = -1

– 1 is an integer.

Consider the two natural numbers 4 and 8.

Then, natural numbers between 4 and 8 are 5, 6, 7.

The successor of a whole number is the number obtained by adding 1 to it.

The largest 3-digit number = 999

Then, its successor = 999 + 1

= 1000

As per the rule, Of the given two natural numbers, the one having more digits is greater.

We know that, sum of two natural numbers is always natural number.

Therefore, natural numbers are closed under addition.

We know that, multiplication of two natural numbers is always natural number.

Therefore, natural numbers are closed under multiplication.

Difference of two natural numbers are not always a natural number.

Therefore, natural numbers are not closed under subtraction.

Let us assume ‘a’ and ‘b’ are the two natural numbers.

Then commutative for natural numbers is a + b = b + a.

Consider the two natural numbers 2 and 4.

Where, a = 2, b = 4

a + b = b + a

2 + 4 = 4 + 2

6 = 6

Zero (0) is the identity for addition of whole numbers.

Consider any whole number i.e. 8.

Then, 8 + 0 = 8

Consider any whole number i.e. 6.

6 × 1 = 6

Zero (0) is a whole number which when added to a whole number, gives the number itself.

We know that, ‘0’ is not a natural number.

Therefore, there is no any natural number which when added to a natural number, gives the number itself.

As per the standard rule, if a whole number is divided by another whole number, which is greater than the first one, the quotient is not equal to zero.

Consider any non-zero whole number i.e. 5

5 is divided by itself = 5/5 = 1

The product of two whole number is always a whole number.

Because, we know that, whole numbers are closed under multiplication.

As per the standard rule, a whole number divided by another whole number greater than 1 never gives the quotient equal to the former.

As per the standard rule, every multiple of a number is greater than or equal to the number.

2 × 1 = 2

2 × 3 = 6

The number of multiples of a given number is infinite.

Because, we know that numbers are infinite.

We know that, 1 is the identity for multiplication of whole numbers

Therefore, any number is multiplied by 1 we get the number itself.

Hence, every number is a multiple of itself.

For example, 1 + 3 = 4 = 4/4 = 1

11 + 13 = 24 = 24/4 = 6

As per the standard rule, if a number divides three numbers exactly, it must divide their sum exactly.

Let us consider one number i.e. 2, it divides 4, 6 and 8.

Then, sum of three numbers = 4 + 6 + 8

= 18 is exactly divisible by 2

Let us consider the number 6, it is actually divisible by 2 and 3, But 6 is not divisible by 12.

As per the rule of divisibility test, a number with 4 or more digits is divisible by 8, if the number formed by the last three digits is divisible by 8.

As per the rule of divisibility test, the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9.

Consider the number 20, it is divisible by 4 but not divisible by 8.

The Highest Common Factor of two or more numbers is lower than their Lowest Common Multiple.

As per the rule, LCM of two or more numbers is divisible by their HCF.

From the question it is given that, LCM of two numbers is 28 and their HCF is 8.

But, 28 is not exactly divide by 8.

Consider the two numbers 2 and 4.

Then, LCM of 2 and 4 is 4.

We know that, 0 is the whole number.

0 is no the successor of another whole number.

For example:-

2 + 3 = 5

2 × 3 = 6

From the above example, we can say that

Consider the two odd numbers 2 and 5.

Then, sum = 2 + 5 = 7 it is an odd number.

Now, difference = 2 – 5 = 3 it also an odd number.

Co-prime number is a set of numbers or integers which have only 1 as their common factor i.e. their highest common factor (HCF) will be 1. Co-prime numbers are also known as relatively prime or mutually prime numbers. It is important that there should be two numbers in order to form co-primes.

The HCF of two numbers is either greater than or equal to the smaller of the numbers.

The LCM of two numbers may be equal to or greater than the larger of the numbers.

10 million = 1 crore

We know that, 1 million = 10 lakh

Then, 10 million = 10 × 10 = 100 lakh = 1,00,00,000

Therefore, 10 million = 1 crore.

10 lakh = 1 million.

1 metre = 1000millimetres

We know that, 1 metre = 100 centimetre

1 centimetre = 10 millimeter

Then, 100 cm = 10 × 100 = 1000 millimetres

1 centimetre = 10 millimetres.

1 kilometre = 10,00,000 millimetres.

We know that, 1 km = 1000 meters.

1 metre = 100 centimetre

1000 metre = 1000 × 100

= 1,00,000 centimetre

1 cm = 10 millimetres

Then, 1,00,000 centimetre = 10 × 1,00,000 = 10,00,000 millimetres

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