Math Help - Finding the remainder

1. Finding 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?

2. 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?

3. 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.

4. 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.

5. 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:
$\frac{N}{P} = Q + \frac{R}{P}$
(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:
$\frac{n}{7} = q + \frac{2}{7}$
Now I want to know what 5n is. Multiply both sides of the equation above by 35 and you'll get
$5n = 35q + 10$

Divide 5n by 7:
\begin{aligned}
\frac{5n}{7} &= \frac{35q + 10}{7} \\
&= \frac{35q}{7} + \frac{10}{7} \\
&= 5q + \frac{10}{7} \\
&= (5q + 1) + \frac{3}{7} \\
\end{aligned}

The quotient would be 5q + 1, and the remainder would be 3.

6. Thank you so much!

7. Another, easier way to do it is to let $n \equiv 2 \pmod{7}$, it then follows that $5n \equiv 10 \equiv 3 \pmod{7}$, thus the remainder is $3$.

8. Originally Posted by Bacterius
Another, easier way to do it is to let $n \equiv 2 \pmod{7}$, it then follows that $5n \equiv 10 \equiv 3 \pmod{7}$, thus the remainder is $3$.
I cannot find the MOD command on my calculator though.

9. It really is the remainder function, usually calculators don't have it. You can emulate $a \mod b$ by dividing $a$ by $b$, taking the fractional part only and multiplying it by $b$. For instance, to get $10 \mod 7$ :

$\frac{10}{7} = 1.428571429 ...$
Take the fractional part which is $0.428571429$, and times it by $7$, you get :
$0.428571429 \times 7 = 3$

10. Originally Posted by Bacterius
Another, easier way to do it is to let $n \equiv 2 \pmod{7}$, it then follows that $5n \equiv 10 \equiv 3 \pmod{7}$, thus the remainder is $3$.
Originally Posted by Bacterius
It really is the remainder function, usually calculators don't have it. You can emulate $a \mod b$ by dividing $a$ by $b$, taking the fractional part only and multiplying it by $b$. For instance, to get $10 \mod 7$ :

$\frac{10}{7} = 1.428571429 ...$
Take the fractional part which is $0.428571429$, and times it by $7$, you get :
$0.428571429 \times 7 = 3$
Oh, ok, but where did you get the 10 from? Especially the 10 in the first post here.
Also, couldn't a shorter way by just looking at 10 mod 7 just be subtracting 7 from 10?

11. If $n \equiv 2 \pmod{7}$ then $\mathbf{5} \times n \equiv \mathbf{5} \times 2 \equiv 10 \equiv 3 \pmod{7}$
This is where the $10$ comes from

Also, couldn't a shorter way by just looking at 10 mod 7 just be subtracting 7 from 10?
That would only work for numbers up to $13$ ... for instance if you substract $7$ from $23$, you get $16$, which isn't exactly the remainder of $23$ divided by $7$, while $23 \mod 7 = 2$ as expected. Then you could argue that repeatedly substracting $7$ is a way to do it, true, but dividing is then more efficient.

12. I totally understand now. Thank you!

13. Hello, Mariolee!

A slightly different approach . . .

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

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

. . Hence: . $n \:=\:7a + 2\:\text{ for some integer }a.$

Then: . $5n \:=\:35a + 10$

. . and: . $\dfrac{5n}{7} \:=\:\dfrac{35a+10}{7} \;=\;5a + 1 + \frac{3}{7}$

Therefore, the remainder is 3.