Number theory question?

Question

if a set of markers is place in rows of 4 each there's 2 left over; in rows of 5 each there's 3 left over; and in rows of 7 each there's 5 left over.

if you call the number of markers x then why does X+2 have to be divisible by 4,5,7.....the part that confuses me is why add 2 ??

Answer 1

To answer your question, as opposed to just solving the problem:

Note that remainder when divided by 4 is 2, which 2 less than 4.

Remainder when divided by 5 is 3, which is also 2 less than 5.

Remainder when divided by 7 is 5, which is again 2 less than 7.

So in each case, if you add 2, it will bring it up to an even multiple.

X + 2 will be divisible by each of 4, 5, and 7.

That makes (X+2) a multiple of 140 = 4 * 5 * 7 (since they have no common factors).

So you can say X = 140 k - 2 for any number k.

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Answer 2

There is  (Z*140) - 2 markers, where Z is any real whole number.

Answer 3

There's 138 markers.