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# Positive integer n leaves a remainder of 4 after division by

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Manager
Joined: 02 Nov 2018
Posts: 50
Re: Positive integer n leaves a remainder of 4 after division by  [#permalink]

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15 Aug 2019, 12:01
Bunuel wrote:
To elaborate more.

Suppose we are told that:
Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 2 after division by 8. What is the remainder that n leaves after division by 12?

The statement "positive integer n leaves a remainder of 4 after division by 6" can be expressed as: $$n=6p+4$$. Thus according to this particular statement $$n$$ could take the following values: 4, 10, 16, 22, 28, 34, 40, 46, 52, 58, 64, ...

The statement "positive integer n leaves a remainder of 2 after division by 8" can be expressed as: $$n=8q+2$$. Thus according to this particular statement $$n$$ could take the following values: 2, 10, 18, 26, 34, 42, 50, 58, 66, ...

The above two statements are both true, which means that the only valid values of $$n$$ are the values which are common in both patterns. For example $$n$$ can not be 16 (from first pattern) as the second formula does not give us 16 for any value of integer $$q$$.

So we should derive general formula (based on both statements) that will give us only valid values of $$n$$.

How can these two statement be expressed in one formula of a type $$n=kx+r$$? Where $$x$$ is divisor and $$r$$ is a remainder.

Divisor $$x$$ would be the least common multiple of above two divisors 6 and 8, hence $$x=24$$.

Remainder $$r$$ would be the first common integer in above two patterns, hence $$r=10$$.

Therefore general formula based on both statements is $$n=24k+10$$. Thus according to this general formula valid values of $$n$$ are: 10, 34, 58, ...

Now, $$n$$ divided by 12 will give us the reminder of 10 (as 24k is divisible by 12).

Hope it helps.

I don't understand the bold red part. I mean I understand the LCM of 6 and 8 is 24 but I find it difficult to understand WHY/HOW this ties to the general formula. Why/how is the LCM relevant? Similarly, why is the remainder the first common integer in the above two patterns..
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Re: Positive integer n leaves a remainder of 4 after division by  [#permalink]

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30 Aug 2019, 11:14
bchekuri wrote:
Positive integer n leaves a remainder of 4 after division by 6 and a remainder of 3 after division by 5. If n is greater than 30, what is the remainder that n leaves after division by 30?

A. 3
B. 12
C. 18
D. 22
E. 28

Well, this is a PS question and options are given. So let's just form the numbers.
30+3=33, 30+12=42 and so on.
33,42,48,52,58.
Now divide these numbers by both 6 and 5 and check if the remainders are satisfied.
58 works.

Hence E
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Re: Positive integer n leaves a remainder of 4 after division by   [#permalink] 30 Aug 2019, 11:14

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