Quotients And Remainders

Number Theory Level 1

If $N$ is an integer that leaves a remainder of 10 upon division by 20,
which of the following can't we uniquely determine?

The remainder of

N

upon division by 2 The remainder of

N

upon division by 3 The remainder of

N

upon division by 4 The remainder of

N

upon division by 5

2 solutions

Chung Kevin
Aug 14, 2016

We are given that $N = 20k + 10$ for some integer $k$ . From that, we get:
$N = 2(10k + 5 ) +0$ , so it will leave a reminder of 0 when divided by 2.
$N = 4(5k + 2 ) +2$ , so it will leave a reminder of 2 when divided by 4.
$N = 5(4k + 2 ) +0$ , so it will leave a reminder of 0 when divided by 5.

However, we only have $N = 3(6k+3) + 2k+1$ , so we the remainder when divided by 3 will depend on the value of $k$ .

Moderator note:

Good explanation! Did you also notice that 2, 4 and 5 are factors of 20? Coincidence?

This is not coincidence. To visualize this, imagine we have several blocks of length 20 and ends with a block of length 10. Then, if we cut it into a factor of 20 (in this case 2, 4, 5) then all the block of length 20 can be grouped and what left for any sets of blocks is the ending block of length 10, so their remaining will also be the same.

Christopher Boo - 4 years, 9 months ago

Zee Ell
Aug 9, 2016

The correct answer is 3, as that is the only option which is not a factor of 20.

We can also look at some examples:

While both 30 and 50 have a remainder of 10 when divided by 20;

30 has a remainder of 0 and 50 has a remainder of 2 when divided by 3.

Quotients And Remainders

2 solutions

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