Page 156 - Ganit Kaushal

Sol. (a) F [ 2 + 3 = 5 (odd) & 2, 3 are prime] (b) T
(c) F [ 2 is even and prime] (d) T
(e) T (f) T
Subjective/Competency Based Questions
Ex 9. In the treasure hunt game, if a jump size of 1 is not allowed, check if these pairs are safe.
(a) 18 and 29 (b) 20 and 55
Sol. (a) It is safe.  18 and 29 are co-prime numbers. Jumpy cannot reach both numbers by any jump size
other than 1 and a jump size of 1 is not allowed.
(b) It is not safe.  With a jump size of 5, Jumpy can reach both 20 and 55.
Ex 10. Identify me!
(a) I am a number < 40, having one of my factors as 7, and the sum of my digits is 8.
(b) I am a number < 100, having two of my factors as 3 and 5, the difference of my digits is 1.
Sol. (a) Numbers with 7 as a factor and < 40 are: 7 × 1 = 7, 7 × 2 = 14, 7 × 3 = 21, 7 × 4 = 28, 7 × 5 = 35.

Out of them, only 35 has a digit sum of 8. \ The required number = 35.
(b) Numbers with 3, 5 as factors and < 100 are: (3 × 5) × 1 = 15, (3 × 5) × 2 = 30, (3 × 5) × 3 = 45,
(3 × 5) × 4 = 60, (3 × 5) × 5 = 75, (3 × 5) × 6 = 90.

Out of them, only 45 has a digit difference of 1. \ The required number = 45.
Ex 11. In the treasure hunt game, Dumpy places two treasures at 69 and 111. A jump size of 1 is not allowed.
Can Jumpy get both the treasures?
Sol. Yes, Jumpy can get both the treasures by choosing a jump size of 3. [ 69 = 3 × 23, and 111 = 3 × 37]

Ex 12. Write 5 prime numbers whose doubling and adding 1 again gives a prime number.
Sol. 5 prime numbers are: 2, 5, 11, 23, 29. [ 2 × 23 + 1 = 47 (prime)] Similarly, we can check for 2, 5,
11, and 29.
Ex 13. Using prime factorisation, check if 3780 is divisible by 756 or not.

Sol. 3780 = 2 × 2 × 3 × 3 × 3 × 5 × 7 and 756 = 2 × 2 × 3 × 3 × 3 × 7. We see that the prime factorisation
of 756 is completely included in that of 3780. \ 3780 is divisible by 756.
Ex 14. Test the divisibility of 415284 by (a) 8, (b) 3, (c) 4 and (d) 6.
Sol. (a) Not divisible by 8. [ The number formed by the last 3 digits of 415284 is (284), which is not
divisible by 8.]
(b) Divisible by 3. [ The sum of the digits of 415284 is 24, which is divisible by 3.]
(c) Divisible by 4. [ The number formed by the last 2 digits of 415284 is (84), which is divisible by 4.]

(d) Divisible by 6. [ The number 415284 is even and the sum of its digits is divisible by 3.]
Miscellaneous Exercise 5.4

There are four options (Q. 1 to Q. 6), out of which only one is correct. Choose the correct option.
1. The number of distinct prime factors of the largest 4-digit number is
(a) 2 (b) 3 (c) 5 (d) 11
2. A number is divisible by both 5 and 6. It may not necessarily be divisible by
(a) 10 (b) 15 (c) 30 (d) 60

3. The sum of the prime factors of 1729 is
(a) 13 (b) 19 (c) 12 (d) 39

154 Mathematics-6