Could somebody answer this Thank you F Least Common Multiple

Could somebody answer this

Thank you

F. Least Common Multiples A least common multiple of two integers a and b is a positive integer c such that (i) a|c and b|C; (ii) if a |x and b|x, then c|x. I Prove: The set of all the common multiples of a and b is an ideal of Z.

Solution

Let m and n be two common multiples of a and b

Then by definition

a|m and b|m and also if a|x and b|x then m|x and similarly for n

Consider m-n

m-n will be divisible by both a and c and also if a|x and b|x, m-n|X

Thus closure property true for subtraction.

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For any r in R, and c in the set of common multiples

rc is also divisible by a and b and also a|x b|x then rc|x

Similarly true for cr.

Thus the set is an ideal of Z.