When the positive integer n is divided by 7, the remainder is 2. What is the remainder when 5n is divided by 7?

How would you set this up?

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- Jul 15th 2010, 11:31 AMMarioleeFinding the remainder
When the positive integer n is divided by 7, the remainder is 2. What is the remainder when 5n is divided by 7?

How would you set this up? - Jul 15th 2010, 11:50 AMwonderboy1953
The first step is to set up 5n/7 = 5/7 + n/7 so to strictly answer your question, except where n is a multiple of 7, then the remainder is n/7. Is this what you're asking about?

- Jul 15th 2010, 12:02 PMMariolee
No. These are the choices:

(A) 2

(B) 3

(C) 4

(D) 5

(E) 6

The answer is 3. I have no freaking clue how they got that. :D - Jul 15th 2010, 12:19 PMwonderboy1953
I think I understand the problem now. When n is 9, then division by 7 leaves a remainder of 2. Now 5 x 9 = 45. Try dividing 45 by 7 and see if you get a remainder of 3.

- Jul 15th 2010, 01:40 PMeumyang
Suppose you divide a number N by P, and you get a quotient Q with a remainder R. This can be rewritten in fraction form:

(remember that 0 <= R < P).

In our case, we are dividing n by 7. You get a quotient q with a remainder of 2, or:

Now I want to know what 5n is. Multiply both sides of the equation above by 35 and you'll get

Divide 5n by 7:

The quotient would be 5q + 1, and the remainder would be 3. - Jul 15th 2010, 03:05 PMMariolee
Thank you so much!

- Jul 15th 2010, 04:36 PMBacterius
Another, easier way to do it is to let , it then follows that , thus the remainder is .

- Jul 15th 2010, 05:21 PMMariolee
- Jul 15th 2010, 05:44 PMBacterius
It really is the remainder function, usually calculators don't have it. You can emulate by dividing by , taking the fractional part only and multiplying it by . For instance, to get :

Take the fractional part which is , and times it by , you get :

- Jul 15th 2010, 08:23 PMMariolee
- Jul 15th 2010, 08:45 PMBacterius
If then

This is where the comes from (Nod)

Quote:

Also, couldn't a shorter way by just looking at 10 mod 7 just be subtracting 7 from 10?

- Jul 15th 2010, 08:53 PMMariolee
I totally understand now. Thank you!

- Jul 16th 2010, 03:34 AMSoroban
Hello, Mariolee!

A slightly different approach . . .

Quote:

When the positive integer is divided by 7, the remainder is 2.

What is the remainder when is divided by 7?

"When is divided by 7, the remainder is 2."

. . Hence: .

Then: .

. . and: .

Therefore, the remainder is 3.