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

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Manager
Joined: 10 Apr 2018
Posts: 182
If the positive integer x is a multiple of 4 and the  [#permalink]

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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.
Manager
Joined: 29 May 2017
Posts: 129
Location: Pakistan
Concentration: Social Entrepreneurship, Sustainability
Re: If the positive integer x is a multiple of 4 and the  [#permalink]

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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.

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.

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

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