Chapter 3 Playing With Numbers Ex 3.5 Solutions
Question - 1 : - Which of the following statements are true?
(a) If a number is divisible by 3, it must be divisible by 9.
(b) If a number is divisible by 9, it must be divisible by 3.
(c) A number is divisible by 18, if it is divisible by both 3 and 6.
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.
(e) If two numbers are co-primes, at least one of them must be prime.
(f) All numbers which are divisible by 4 must also be divisible by 8.
(g) All numbers which are divisible by 8 must also be divisible by 4.
(h) If a number exactly divides two numbers separately, it must exactly divide their sum.
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.
Answer - 1 : -
(a) False
6 is divisible by 3, but not by 9.
(b) True, as 9 = 3 × 3
Therefore, if a number is divisible by 9, then it will also be divisible by
3.
(c) False
30 is divisible by 3 and 6 both, but it is not divisible by 18.
(d) True, as 9 × 10 = 90
Therefore, if a number is divisible by 9 and 10 both, then it will also be divisible by 90.
(e) False
15 and 32 are co-primes and also composite.
(f) False
12 is divisible by 4, but not by 8.
(g) True, as 8 = 2 × 4
Therefore, if a number is divisible by 8, then it will also be divisible by 2 and 4.
(h) True
2 divides 4 and 8 as well as 12. (4 + 8 = 12)
(i) False
2 divides 12, but does not divide 7 and 5.
Question - 2 : - Here are two different factor trees for 60. Write the missing numbers.
(a)
(b)
Answer - 2 : - (a) As 6 = 2 × 3 and 10 = 5 × 2
(b) As 60 = 30 × 2, 30 = 10 × 3, and 10 = 5 × 2
Question - 3 : - Which factors are not included in the prime factorization of a composite number?
Answer - 3 : -
1 and the number itself are not included in the prime factorisation of a composite number.
Question - 4 : - Write the greatest 4-digit number and express it in terms of its prime factors.
Answer - 4 : -
Greatest four-digit number = 9999
9999 = 3 × 3 × 11 × 101
Question - 5 : - Write the smallest 5-digit number and express it in the form of its prime factors.
Answer - 5 : -
Smallest five-digit number = 10,000
10000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 × 5
Question - 6 : - Find all prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.
Answer - 6 : - 
1729 = 7 × 13 × 19
13 – 7 = 6
19 – 13 = 6
Hence, the difference between two consecutive prime factors is 6.
Question - 7 : - The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.
Answer - 7 : -
2 × 3 × 4 = 24, which is divisible by 6
9 × 10 × 11 = 990, which is divisible by 6
20 × 21 × 22 = 9240, which is divisible by 6
Question - 8 : - The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.
Answer - 8 : -
3 + 5 = 8, which is divisible by 4
15 + 17 = 32, which is divisible by 4
19 + 21 = 40, which is divisible by 4
Question - 9 : - In which of the following expressions, prime factorization has been done?
(a) 24 = 2 × 3 × 4 (b) 56 = 7 × 2 × 2 × 2
(c) 70 = 2 × 5 × 7 (d) 54 = 2 × 3 × 9
Answer - 9 : -
(a) 24 = 2 × 3 × 4
Since 4 is composite, prime factorisation has not been done.
(b) 56 = 7 × 2 × 2 × 2
Since all the factors are prime, prime factorisation has been done.
(c) 70 = 2 × 5 × 7
Since all the factors are prime, prime factorisation has been done.
(d) 54 = 2 × 3 × 9
Since 9 is composite, prime factorisation has not been done.
Question - 10 : - Determine if 25110 is divisible by 45.
[Hint: 5 and 9 are co-prime numbers. Test the divisibility of the number by 5 and 9].
Answer - 10 : -
45 = 5 × 9
Factors of 5 = 1, 5
Factors of 9 = 1, 3, 9
Therefore, 5 and 9 are co-prime numbers.
Since the last digit of 25110 is 0, it is divisible by 5.
Sum of the digits of 25110 = 2 + 5 + 1 + 1 + 0 = 9
As the sum of the digits of 25110 is divisible by 9, therefore, 25110 is divisible by 9.
Since the number is divisible by 5 and 9 both, it is divisible by 45.