(i) If a number is divisible by 3, it must be divisible by 9.

False, as any number following criteria of 9n + 3 or 9n + 6 violates the statement.

For example, 6, 12,…

(ii) If a number is divisible by 9, it must by divisible by 3.

True, as 9 is multiple of 3.

Hence, every number which is divisible by 9 must be divisible by 3.

(iii) If a number is divisible by 4, it must by divisible by 8.

False, as any number following criteria of 8n + 4 violates the statement.

For example, 4, 12,20,….

(iv) If a number is divisible by 8, it must be divisible by 4.

True, as 8 is multiple of 4.

Hence, every number which is divisible by 8 must be divisible by 4.

(v) A number is divisible by 18, if it is divisible by both 3 and 6.

False, for example 48, which is divisible to both 3 and 6 but not divisible with 18

(vi) If a number is divisible by both 9 and 10, it must be divisible by 90.

True, as 90 is the GCD of 9 and 10.

Hence, every number which is divisible by both 9 and 10, it must be divisible by 90.

(vii) If a number exactly divides the sum of two numbers, it must exactly divide the numbers separately.

False, for example 6 divides 30, but 6 divides none of 13 and 17 as both are prime numbers.

(viii) If a number divides three numbers exactly, it must divide their sum exactly.

True, if x,y and z are three numbers, where each of x, y and z is divided by a number (say s), then (x+y+z) is divided by s

(ix) If two number are co-prime, at least one of them must be a prime number.

False, as 16 and 21 are co prime but none of them is prime.

(x) The sum of two consecutive odd numbers is always divisible by 4.

True.