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.

To do:

We have to find whether the given statement is true or false.

Solution :

(i) We know that,

If a number is divisible by another number, then it is divisible by its factors.

Therefore,

If a number is divisible by 9 it is divisible by 3 but the converse is not necessarily true.

For example,

12 and 15 are divisible by 3 but not by 9.

Therefore, if a number is divisible by 3 it may not be divisible by 9.

The given statement is false.

(ii) We know that,

If a number is divisible by another number, then it is divisible by its factors.

Therefore,

If a number is divisible by 9 it is divisible by 3

The given statement is true.

(iii) We know that,

If a number is divisible by another number, then it is divisible by its factors.

Therefore,

If a number is divisible by 8 it is divisible by 4 but the converse is not necessarily true.

For example,

12 and 20 are divisible by 4 but not by 8.

Therefore, if a number is divisible by 4 it may not be divisible by 8.

The given statement is false. 

(iv) We know that,

If a number is divisible by another number, then it is divisible by its factors.

Therefore,

If a number is divisible by 8 it is divisible by 4.

The given statement is true. 

(v) A number is divisible by 18 if it is divisible by both 9 and 2.

The given statement is false. 

(vi) A number is divisible by 90 if it is divisible by both 9 and 10.

The given statement is true. 

(vii) $10+30=40$ is divisible by 4 but both 10 and 30 are not divisible by 4. 

The given statement is false. 

(viii) Let the three numbers be $4, 6$ and $8$

2 divides 4, 6 and 8 exactly.

$4+6+8=18$

2 divides 18 exactly.

The given statement is true.

(ix) When two numbers have no common factors other than 1, then the numbers are co prime.

8 and 9 are co-primes but both 8 and 9 are not prime numbers.

The given statement is false. 

(x) Let $x$ be an odd number.

This implies,

$x+1$ is an even number.

The next odd number is $x+2$

Sum of the consecutive odd numbers $= x+x+2$

$=2x+2$

$(2x+2)\div4= \frac{2x+2}{4}$

$=\frac{x+1}{2}$

$x+1$ is divisible by 2.

This implies,

$2x+2$ is divisible by 4

Therefore, the sum of two consecutive odd numbers is always divisible by 4.

The given statement is true.