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Re: What is the remainder when positive integer x is divided by [#permalink]

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05 Mar 2013, 09:47

1

This post was BOOKMARKED

manimgoindowndown wrote:

What is the remainder when positive integer x is divided by 3? 1) When x is divided by 6, the remainder is 2 2) When x is divided by 15, the remainder is 2

From F.S 1, we have that x = 6k+2, where k is a non-negative integer constant. Required to find x = 3p+r, where p= again a non-negative integer constant. We can see that for some value, 6k =\(3*(2k)\) = 3p. Thus, the remainder when divided by 3 will also be 2. Sufficient.

Similarly, from F.S 2 , we have that x = 15t+2. Just as above, for some integer, 15t = \(3*(5t)\) = 3p. Thus, the remainder is 2.Sufficient.

Thanks even though it's my off day I'm going to run through that thread. Thanks for sorting the problems by difficulty. Your work and its contribution has been immense to my preparation Bunuel
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If \(x\) and \(y\) are positive integers, there exist unique integers \(q\) and \(r\), called the quotient and remainder, respectively, such that \(y =divisor*quotient+remainder= xq + r\) and \(0\leq{r}<x\).

For example, when 15 is divided by 6, the quotient is 2 and the remainder is 3 since \(15 = 6*2 + 3\).

Notice that \(0\leq{r}<x\) means that remainder is a non-negative integer and always less than divisor.

This formula can also be written as \(\frac{y}{x} = q + \frac{r}{x}\).
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These types of 'remainder' questions are almost always easily solved by TESTing VALUES.

We're told that X is a positive integer. We're asked for the remainder when X is divided by 3.

Fact 1: When X is divided by 6, the remainder is 2

I'm going to list out the first few integers that fit this description:

X = 2, 8, 14, 20, 26, 32, etc....

The pattern here is that each number is "6 more" than the one before it. Now, let's see what happens when we use these values in the question:

IF... X = 2 2/3 = 0 remainder 2

X = 8 8/3 = 2 remainder 2

X = 14 14/3 = 4 remainder 2

X = 20 20/3 = 6 remainder 2 Etc.

The pattern here is clear (and you could probably name the next few "results" without doing any calculations at all). The remainder is ALWAYS 2. Fact 1 is SUFFCIENT

Fact 2: When X is divided by 15, the remainder is 2

Here are the first few terms that fit this Fact:

X = 2, 17, 32, 47, etc.

IF.... X = 2 2/3 = 0 remainder 2

X = 17 17/3 = 5 remainder 2

X = 32 32/3 = 10 remainder 2

X = 47 47/3 = 15 remainder 2 Etc.

Just as in Fact 1, we have a clear pattern here. The answer is ALWAYS 2. Fact 2 is SUFFICIENT

Re: What is the remainder when positive integer x is divided by [#permalink]

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20 Mar 2016, 13:11

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