Page 151 - Ganit Kaushal

Ex 3. Using prime factorisation, check the divisibility of 168 by 12. (NCERT)

Sol. Prime factorisation of 168 = 2 × 2 × 2 × 3 × 7; prime factorisation of 12 = 2 × 2 × 3.
Now, prime factorisation of 12 is completely included in the prime factorisation of 168.
Therefore, 168 is divisible by 12.
Ex 4. Find the prime factorisation of the following numbers: 320, 141, 1728 (NCERT)
Sol. Prime factorisation of 320 = 10 × 32 = 2 × 5 × 2 × 16 = 2 × 5 × 2 × 2 × 2 × 2 × 2
Prime factorisation of 141 = 3 × 47

Prime factorisation of 1728 = 4 × 432 = 2 × 2 × 4 × 108 = 2 × 2 × 2 × 2 × 4 × 27
= 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3
Ex 5. The prime factorisation of a number has one 2, two 3s, and one 11. What is the number? (NCERT)
Sol. The required number = 2 × 3 × 3 × 11 = 18 × 11 = 198.
Ex 6. Find three prime numbers, all less than 30, whose product is 1955. (NCERT)
Sol. 1955 = 5 × 391 = 5 × 17 × 23
\ 5, 17, 23 are three prime numbers, all less than 30, whose product is 1955.

Ex 7. Find the prime factorisation of these numbers without multiplying first (NCERT)
(a) 56 × 25 (b) 1000 × 81
Sol. (a) 56 × 25 = 7 × 8 × 5 × 5 = 7 × 2 × 2 × 2 × 5 × 5
(b) 1000 × 81 = 10 × 10 × 10 × 9 × 9 = 5 × 2 × 5 × 2 × 5 × 2 × 3 × 3 × 3 × 3
Ex 8. What is the smallest number whose prime factorisation has: (NCERT)

(a) three different prime numbers? (b) four different prime numbers?
Sol. (a) The smallest number whose prime factorisation has three different prime factors = 2 × 3 × 5 = 30
(b) The smallest number whose prime factorisation has four different prime factors = 2 × 3 × 5 × 7 = 210
Ex 9. Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify
your answer. (NCERT)
(a) 30 and 45 (b) 57 and 85
Sol. (a) 30 and 45 are not co-prime.  3 and 5 are the common factors of 30 and 45.
(b) 57 = 19 × 3 and 85 = 17 × 5. So, they are co-prime. ( No common factor)
Ex 10. Is the first number divisible by the second? Use prime factorisation. (NCERT)
(a) 225 and 27 (b) 96 and 24

Sol. (a) Prime factorisation of 225 = 5 × 5 × 3 × 3 and prime factorisation of 27 = 3 × 3 × 3.
The prime factors of 27 are not included completely in the prime factors of 225. Therefore, 225 is
not divisible by 27.
(b) Prime factorisation of 96 = 3 × 2 × 2 × 2 × 2 × 2 and prime factorisation of 24 = 3 × 2 × 2 × 2.
The prime factors of 24 are included completely in the prime factors of 96. Therefore, 96 is divisible
by 24.
Ex 11. The first number has prime factorisation 2 × 3 × 7 and the second number has prime factorisation
3 × 7 × 11. Are they co-prime? Does one of them divide the other? (NCERT)
Sol. No, since 3 × 7 = 21 is common in the prime factorisation of both the numbers. So, they are not co-primes.
No, since the prime factorisation of one number does not appear in the prime factorisation of the other
completely. So, neither one divides the other.

Prime Time 149