Expert: Paul Klarreich - 9/21/2008

Question
Hey Paul

(page 523)
When 15 is divided by the positive integer k, the remainder is 3. For how many different values of k is this true?
a) One b) Two c)Three d) Four e) Five

I don't get why does 'the remainder is 3' mean.
Does it mean like this? : 15/k = x+ 3? If this is correct, how do i solve x to find how many different values for k? (x is the number 15/k equals plus remainder 3)

Thanks in advance

Answer
Questioner:   Emily
Category:  Advanced Math
Private:  No
 
Subject:  How many different values for k?
Question:  Hey Paul

(page 523)  << what is that?
When 15 is divided by the positive integer k, the remainder is 3. For how many different values of k is this true?
a) One b) Two c)Three d) Four e) Five

I don't get why does 'the remainder is 3' mean.
Does it mean like this? : 15/k = x+ 3? If this is correct, how do i solve x to find how many different values for k? (x is the number 15/k equals plus remainder 3)

Thanks in advance
.......................................
Hi, Emily,

The 'division algorithm' for integers says:

A = k q + r, where   r is between  0 and k-1


A is called the dividend,
k is called the divisor,
q is called the quotient
r is called the remainder.

Here you have:

15 = kq + 3.

So the divisor cannot be smaller than 4.

So try some values

15 divided by  4 gives r = 3, because  15 = 4(3) + 3  [good]
15 divided by  5 gives r = 0  [no good]
15 divided by  6 gives r = 3  [good]
15 divided by  7 gives r = 1  [no good]
15 divided by  8 gives r = 7  [no good]
15 divided by  9 gives r = 6  [no good]
15 divided by 10 gives r = 5  [no good]
15 divided by 11 gives r = 4  [no good]
15 divided by 12 gives r = 3  [good]
15 divided by 13 gives r = 2  [no good]
15 divided by 14 gives r = 1  [no good]
15 divided by 15 gives r = 0  [no good]

15 divided by something larger than 15 gives r = 15  [no good]

Looks like the answer is 3 possible divisors.