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If the positive integer x is a multiple of 4 and the

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If the positive integer x is a multiple of 4 and the  [#permalink]

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New post Updated on: 07 Sep 2018, 15:32
Hi,

Well after doing a couple of questions in this topic i have drawn some conlusions that can come handy for this topic

1. if A is divisible by B, and B is divisible by C , then \(\frac{A}{C}= integer\) or we say A is divisible by C.
2. if A , B are natural nos then such that A is divisible by B , and B is divisible by A then A=B
3. if A is divisible by d, and B is divisible by d , then \(\frac{A+B}{d}= integer\) , \(\frac{A- B}{d}= integer\), \(\frac{ B+A}{d}= integer\),\(\frac{ B-A}{d}= integer\).
4. if A is divisible by C, and B is divisible by D , then \(\frac{AB}{CD}= integer\)
5. \(\frac{a^n}{a+1}\) leaves Remainder = a if n is odd, else 1 if n is even
6. Always \(\frac{(a+1)^n}{a}\) remainder always 1.

So in this question we use the property 4. and we can say that \(\frac{xy}{24} =integer\)

So xy must have 8, 12 as its mulitples


Probus

Originally posted by Probus on 26 Aug 2018, 22:42.
Last edited by Probus on 07 Sep 2018, 15:32, edited 1 time in total.
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Re: If the positive integer x is a multiple of 4 and the  [#permalink]

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New post 07 Sep 2018, 15:24
Bunuel wrote:
SOLUTION

If the positive integer x is a multiple of 4 and the positive integer y is a multiple of 6, then xy must be a multiple of which of the following?

I. 8
II. 12
III. 18

(A) II only
(B) I and II only
(C) I and III only
(D) II and III only
(E) I, II, and III

First of all notice that we are asked: "\(xy\) MUST be a multiple of which of the following", not COULD be a multiple.

\(x\) is a multiple of 4 --> \(x=4m\), for some positive multiple \(m\), so \(x\) could be: 4, 8, 12, ...
\(y\) is a multiple of 6 --> \(y=6n\), for some positive multiple \(n\), so \(y\) could be: 6, 12, 18, ...

So, \(xy=(4m)*(6n)=24mn\), hence \(xy\) is in any case a multiple of 24, which means it must be a multiple of 8 and 12, but not necessarily of 18. For example the least value of \(x\) is 4 and the least value of \(y\) is 6, so the least value of \(xy\) is 24, which is a multiple of both 8 and 12, but not 18.

Answer: B.

Hope it's clear.


Hi....a silly question:
"if 6 is a multiple of a number then 12 is also a multiple of that number" True or false?

x: 6n--> 6,12,18,24
since 18 is not a multiple of 12, i say false. but the book answer is true.

can you please explain?
(this question is from the O-levels math book)
GMAT Club Bot
Re: If the positive integer x is a multiple of 4 and the   [#permalink] 07 Sep 2018, 15:24

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