Which of the following statements are true?

Question:

Which of the following statements are true?

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

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

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

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

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

(vi) If a number 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.

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

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

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

Solution:

(i) False

Every number with the structures (9n + 3) or (9n + 6) is divisible by 3 but not by 9. Example: 3, 6, 12 etc.

(ii) True

(iii) False

Every number with the structure (8n + 4) is divisible by 4 but not by 8. Example: 4, 12, 20 etc.

(iv) True

(v) False

Example: 24 is divisible by both 3 and 6 but it is not divisible by 18.

(vi) True

(vii) False

Example: 5 divides 10, which is a sum of 3 and 7. However, it neither divides 3 nor 7.

(viii) True

(ix) False

Example: 4 and 9 are co-prime numbers but both are composite numbers too.

(x) True