a, b, c and d are non-zero integers such that (ab) divides (cd). If a and c are coprime, then which one of the following is correct?a)a is a factor of cb)a is a factor of bc)a is a factor of dd)d is a factor of aCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Solution:

Given, (ab) divides (cd) and a and c are coprime.

We know that if a and b are coprime, then (a,b) = 1.

Therefore, (a,c) = 1.

Let (cd)/(ab) = k, where k is an integer.

We can write c = ka' and d = kb' where a' and b' are integers.

Substituting these values in (ab) divides (cd), we get,

ab divides kab'b

Dividing both sides by b, we get,

a divides kab'

Since (a,c) = 1, a does not divide k.

Therefore, a must divide b.

Hence, option A is correct which states that a is a factor of b.

Conclusion:

Therefore, we can conclude that if (ab) divides (cd) and a and c are coprime, then a must be a factor of b.

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