If a number exactly divides two numbers separately, it must exactly divide their sum.

True

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False

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Solution

The correct option is A True
Consider a common factor of two numbers to check the divisibility of their sum:
If a number divides another number exactly, then it is considered to be a factor of the number.
Let the two numbers in consideration be $A x$ and $B x$ .
Let $x$ be the number that divides both these numbers exactly.
The sum of these two numbers is $A x + B x = x (A + B)$ .
Hence $x$ is a factor of the sum of the two numbers consequently dividing the sum of two numbers exactly.
For eg: $7$ divides the numbers $21$ and $49$ exactly.
The sum of these two numbers is $70$
$7$ divides $70$ exactly.
Hence the given statement is true.

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Similar questions

If a number exactly divides two numbers separately, it must exactly divide their sum.

Question 80

In the following question, state whether the given statement is true (T) or false (F).
If a number exactly divides the sum of three numbers, it must exactly divide the numbers separately.

Q. Write (T) for true and (F) for false against each of the following statements:
(i) If a number is divisible by 4, it must be divisible by 8.
(ii) If a number is divisible by 8, it must be divisible by 4.
(iii) If a number divides the sum of two numbers exactly, it must exactly divide the numbers separately.
(iv) If a number is divisible by both 9 and 10, it must be divisible by 90.
(v) A number is divisible by 18 if it is divisible by both 3 and 6.
(vi) If a number is divisible by 3 and 7, it must be divisible by 21.
(vii) The sum of two consecutive odd numbers is always divisible by 4.
(viii) If a number divides two numbers exactly, it must divide their sum exactly.

Question 79

In the following question, state whether the given statement is true (T) or false (F).
If a number divides three numbers exactly, it must divide their sum exactly.

Q. 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.

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