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If \(a\) and \(b\) are positive integers, is \(a\) a multiple of \(b?\)

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If \(a\) and \(b\) are positive integers, is \(a\) a multiple of \(b?\)

by M7MBA » Fri Jul 24, 2020 7:42 am

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If \(a\) and \(b\) are positive integers, is \(a\) a multiple of \(b?\)

(1) Every distinct prime factor of \(b\) is also a distinct prime factor of \(a.\)

(2) Every factor of \(b\) is also a factor of \(a.\)

[spoiler]OA=B[/spoiler]

Source: Princeton Review
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Re: If \(a\) and \(b\) are positive integers, is \(a\) a multiple of \(b?\)

by dashingnightmare » Fri Jul 24, 2020 10:49 pm

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Is A , a multiple of b
(1) Every distinct prime factor of b is also a distinct prime factor of a
lets take a =2,b=4,
but is not a multiple of b

(2) Every factor of b is also a factor of a:
We cannot take a=2,b=4 and every case will suffice, hence enough

So,B
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Re: If \(a\) and \(b\) are positive integers, is \(a\) a multiple of \(b?\)

by hahahehe » Sat Jul 25, 2020 4:42 am

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let's evaluate statement 1.
Let B's distinct prime factors be: 2, 3, 5
Now B can be 2*3*5, or it can be 2*2*3*5.. or basically anything in the form (2^x)*(3^y)*(5^z)

Now, we know a has all the prime factors that B has, as its own prime factors, and it can also have more.
A can be = 2*3*5, or 2*3*5*7, or 2*3*5*11

But is A a multiple of B? We cannot say, because if any of the prime factor's power in A is less than that in B, A won't be a multiple of B.
Example, B = 2*3*3*5*5, A = 2*3*5*7. Here, A has all prime factors that are in B, but it's not a multiple of B

Statement 2: If every factor of B is factor of A, we surely know A is multiple of B. Since, a number itself is also a factor of itself. 6 is a factor of 6, 10 is a factor of 10.. And if B is a factor of A, then A is definitely a multiple of B

So we just need statement 2 to confirm that A is multiple of B
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