Playing With Numbers
Exercise 2.1 (Factors and Multiples)
Question 1 (Refer Book  Mathematics Class VI R.D. Sharma) 
Solution: 
Factor – A factor of a number is an exact divisor of that number. Example: Each of the number 1, 3, 5 and 15 is a factor of 15. However, none of the numbers 2, 4, 6, 7, 8, 9, 10, 11, 12, 13 and 14 is a factor of 12.
Multiple – A multiple of a number is a number obtained by multiplying it by a natural number. Example: If we multiply 4 by 1, 2, 3, 4, 5, 6, ….., we get 4x1=4, 4x2=8, 4x3=12, 4x4=16, 4x5=20, 4x6=24 ……. Thus, 4, 8, 12, 16, 20, 24 ……… are multiples of 4. 
Question 2 (Refer Book  Mathematics Class VI R.D. Sharma) 
Solution: 
i. 60 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 are the numbers which exactly divides the given number. Therefore, Factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
ii. 76 1, 2, 4, 19, 38 and 76 are the numbers which exactly divides the given number. Therefore, Factors of 76 are 1, 2, 4, 19, 38 and 76.
iii. 125 1, 5, 25 and 125 are the numbers which exactly divides the given number. Therefore, Factors of 125 are 1, 5, 25 and 125.
iv. 729 1, 3, 9, 27, 81, 243 and 729 are the numbers which exactly divides the given number. Therefore, Factors of 729 are 1, 3, 9, 27, 81, 243 and 729.

Question 3 (Refer Book  Mathematics Class VI R.D. Sharma) 
Solution: 
i. 25 In order to find first five multiples of 25, we multiply the given number by 1, 2, 3, 4 and 5 respectively. We have, 25x1 = 25 25x2 = 50 25x3 = 75 25x4 = 100 25x5 = 125 Hence, first five multiples of 25 are 25, 50, 75, 100 and 125 respectively.
ii. 35 In order to find first five multiples of 35, we multiply the given number by 1, 2, 3, 4 and 5 respectively. We have, 35x1 = 35 35x2 = 70 35x3 = 105 35x4 = 140 35x5 = 175 Hence, first five multiples of 35 are 35, 70, 105, 140 and 175 respectively.
iii. 45 In order to find first five multiples of 45, we multiply the given number by 1, 2, 3, 4 and 5 respectively. We have, 45x1 = 45 45x2 = 90 45x3 = 135 45x4 = 180 45x5 = 225 Hence, first five multiples of 45 are 45, 90, 135, 180 and 225 respectively.
iv. 40 In order to find first five multiples of 40, we multiply the given number by 1, 2, 3, 4 and 5 respectively. We have, 40x1 = 40 40x2 = 80 40x3 = 120 40x4 = 160 40x5 = 200 Hence, first five multiples of 40 are 40, 80, 120, 160 and 200 respectively. 
Question 4 (Refer Book  Mathematics Class VI R.D. Sharma) 
Solution: 
i. 15625 15625 ÷ 15 = 1041.67 As the given number is not exactly divisible by 15, therefore 15 is not the factor of 15625.
ii. 123015 123015 ÷ 15 = 8201 As the given number is exactly divisible by 15, therefore 15 is the factor of 123015.

Question 5 (Refer Book  Mathematics Class VI R.D. Sharma) 
Solution: 
i. 21063 21063 ÷ 21 = 1003 As the given number is exactly divisible by 21, therefore 21 is the factor of 21063.
ii. 20163 20163 ÷ 21 = 960.14 As the given number is not exactly divisible by 21, therefore 21 is not the factor of 20163. 
Question 6 (Refer Book  Mathematics Class VI R.D. Sharma) 
Solution: 
Divisibility Rule of 11 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.
i. 1111 Sum of digits at odd places from the right = 1+1 = 2 Sum of digits at even places from the right = 1+1 = 2 Difference = 2 – 2 = 0 The difference is 0. Therefore, 1111 is divisible by 11
ii. 11011 Sum of digits at odd places from the right = 1+0+1 = 2 Sum of digits at even places from the right = 1+1 = 2 Difference = 2 – 2 = 0 The difference is 0. Therefore, 11011 is divisible by 11
iii. 110011 Sum of digits at odd places from the right = 1+0+1 = 2 Sum of digits at even places from the right = 1+0+1 = 2 Difference = 2 – 2 = 0 The difference is 0. Therefore, 110011 is divisible by 11
iv. 1100011 Sum of digits at odd places from the right = 1+0+0+1 = 2 Sum of digits at even places from the right = 1+0+1 = 2 Difference = 2 – 2 = 0 The difference is 0. Therefore, 1100011 is divisible by 11

Question 7 (Refer Book  Mathematics Class VI R.D. Sharma) 
Solution: 
Divisibility Rule of 5 To check the divisibility of a number by 5, the rule is, a number which has either 0 or 5 in its ones place is divisible by 5.
i. 55 The given number has 5 in its ones place. Therefore ‘55’ is divisible by 5.
ii. 555 The given number has 5 in its ones place. Therefore, ‘555’ is divisible by 5.
iii. 5555 The given number has 5 in its ones place. Therefore, ‘5555’ is divisible by 5.
iv. 50005 The given number has 5 in its ones place. Therefore, ‘50005’ is divisible by 5. 
Question 8 (Refer Book  Mathematics Class VI R.D. Sharma) 
Solution: 
No, there no natural number having no factor at all. 
Question 9 (Refer Book  Mathematics Class VI R.D. Sharma) 
Solution: 
Numbers between 1 and 100 having exactly three factors are 4, 9, 25 and 49. 
Question 10 (Refer Book  Mathematics Class VI R.D. Sharma) 
Solution: 
Even Numbers – 42 and 144 Odd Numbers – 89 and 321 