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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.

Solution:

(a) If a number is divisible by 3, it must be divisible by 9 is a false statement because 9 is not a factor of 3. Example: 6 is divisible by 3 but not divisible by 9.
(b) If a number is divisible by 9, it must be divisible by 3 is a true statement as 9 = 3 × 3 i.e, 3 is a factor o 9. Example: 36, 45, 54 are divisible by both 9 and 3.
(c) A number is divisible by 18, if it is divisible by both 3 and 6 is a false statement because 30 is divisible by 3 and 6 but not divisible by 18.
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90 is a true statement because if a number is divisible by two co-prime numbers then it is divisible by its product also. Here, 9 and 10 are co-prime numbers and 90 can be expressed as 90 = 10 × 9. Example: 900 is divisible by both 9 and 10 and hence, by 90.
(e) If two numbers are co-primes, at least one of them must be prime is a false statement as any two numbers having HCF as 1 are known as co-prime numbers. The numbers necessarily do not have to be prime. Example: 4 and 9 are co-primes as their HCF is equal to 1 but none of the numbers are prime.
(f) All numbers which are divisible by 4 must also be divisible by 8 is a false statement because 8 is not a factor of 4. Example: 12 is divisible by 4 but not divisible by 8.
(g) All numbers which are divisible by 8 must also be divisible by 4 is a true statement because as 8 = 4 × 2 i.e, 4 is a factor of 8. Example: 48, 64, 80 are divisible by both 8 and 4.
(h) If a number exactly divides two numbers separately, it must exactly divide their sum is a true statement because if two numbers are divisible by a number, their sum is also divisible by the number. Example: 3 divides 6 and 9 it also divides 15. (6 + 9 = 15)
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately is a false statement because 3 divides 18 but not divides 14 and 4 separately.

NCERT Solutions for Class 6 Maths Chapter 3 Exercise 3.5 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.

Summary:

The statements (b), (d), (g), (h) are true and the rest of the statements (a), (c), (e), (f) and (i) are false.

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