Page 152 - Ganit Kaushal

Ex 12. Guna says, “Any two prime numbers are co-prime”. Is he right? (NCERT)

Sol. Yes, he is right. ( Their prime factorisations have nothing in common.)
Ex 13. Find numbers between 330 and 340 that are divisible by 4. Also, find numbers between 1730 and
1740, and 2030 and 2040, that are divisible by 4. What do you observe? (NCERT)

Sol. By hit and trial method, numbers that are divisible by 4 are: 332, 336; 1732, 1736; 2032, 2036; we
observe that the last two digits are divisible by 4.

Ex 14. Is 8536 divisible by 4? (NCERT)
Sol. Yes, because the last two digits of the number 8536, that is, 36 is divisible by 4.
Ex 15. Consider these statements: (NCERT)
(a) Only the last two digits matter when deciding if a given number is divisible by 4.
(b) If the number formed by the last two digits is divisible by 4, then the original number is divisible
by 4.
(c) If the original number is divisible by 4, then the number formed by the last two digits is divisible
by 4.
Do you agree?
Sol. (a) Yes (b) Yes (c) Yes

Ex 16. Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.
(NCERT)
Sol. A palindromic number is a number that reads the same forward and backward.
The smallest 4-digit palindrome that is divisible by 4 = 2112
[ Last 2 digits of the number is divisible by 4.]
The largest 4-digit palindrome that is divisible by 4 = 8888.
Ex 17. Explore and find out if each statement is always true, sometimes true or never true. You can give
examples to support your reasoning. (NCERT)
(a) The sum of two even numbers gives a multiple of 4.
(b) The sum of two odd numbers gives a multiple of 4.

Sol. (a) Sometimes true.  2 + 2 = 4 (A multiple of 4) and 2 + 4 = 6 (Not a multiple of 4)
(b) Sometimes true.  3 + 3 = 6 (Not a multiple of 4) and 3 + 5 = 8 (A multiple of 4)
Ex 18. The teacher asked if 14560 is divisible by all of 2, 4, 5, 8 and 10. Guna checked for divisibility of
14560 by only two of these numbers and then declared that it was also divisible by all of them. What
could those two numbers be? (NCERT)
Sol. The two numbers are 8 and 5. Since the last 3 digits of 14560 is (560) which is divisible by 8 and the
last digit of 14560 is (0), so it is divisible by 5. Now if it is divisible by 8, it is divisible by 2 and 4 as
well. Further, if it is divisible by 5 and 2, then it is divisible by 5 × 2 = 10. ( 2 and 5 are co-prime)
Ex 19. Write two numbers whose product is 10000. The two numbers should not have 0 as the units digit.
(NCERT)
Sol. Prime factorisation of 10000 = 10 × 10 × 10 × 10

= 5 × 2 × 5 × 2 × 5 × 2 × 5 × 2
= 5 × 5 × 5 × 5 × 2 × 2 × 2 × 2 ( we can write it in any order)
= 625 × 16
Thus 625 and 16 are the two numbers that do not have 0 as the units digit but their product is 10000.

150 Mathematics-6