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?
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:
$\displaystyle \frac{N}{P} = Q + \frac{R}{P}$
(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:
$\displaystyle \frac{n}{7} = q + \frac{2}{7}$
Now I want to know what 5n is. Multiply both sides of the equation above by 35 and you'll get
$\displaystyle 5n = 35q + 10$
Divide 5n by 7:
$\displaystyle \begin{aligned}
\frac{5n}{7} &= \frac{35q + 10}{7} \\
&= \frac{35q}{7} + \frac{10}{7} \\
&= 5q + \frac{10}{7} \\
&= (5q + 1) + \frac{3}{7} \\
\end{aligned}$
The quotient would be 5q + 1, and the remainder would be 3.
It really is the remainder function, usually calculators don't have it. You can emulate $\displaystyle a \mod b$ by dividing $\displaystyle a$ by $\displaystyle b$, taking the fractional part only and multiplying it by $\displaystyle b$. For instance, to get $\displaystyle 10 \mod 7$ :
$\displaystyle \frac{10}{7} = 1.428571429 ...$
Take the fractional part which is $\displaystyle 0.428571429$, and times it by $\displaystyle 7$, you get :
$\displaystyle 0.428571429 \times 7 = 3$
If $\displaystyle n \equiv 2 \pmod{7}$ then $\displaystyle \mathbf{5} \times n \equiv \mathbf{5} \times 2 \equiv 10 \equiv 3 \pmod{7}$
This is where the $\displaystyle 10$ comes from
That would only work for numbers up to $\displaystyle 13$ ... for instance if you substract $\displaystyle 7$ from $\displaystyle 23$, you get $\displaystyle 16$, which isn't exactly the remainder of $\displaystyle 23$ divided by $\displaystyle 7$, while $\displaystyle 23 \mod 7 = 2$ as expected. Then you could argue that repeatedly substracting $\displaystyle 7$ is a way to do it, true, but dividing is then more efficient.Also, couldn't a shorter way by just looking at 10 mod 7 just be subtracting 7 from 10?
Hello, Mariolee!
A slightly different approach . . .
When the positive integer $\displaystyle n$ is divided by 7, the remainder is 2.
What is the remainder when $\displaystyle 5n$ is divided by 7?
"When $\displaystyle n$ is divided by 7, the remainder is 2."
. . Hence: .$\displaystyle n \:=\:7a + 2\:\text{ for some integer }a.$
Then: .$\displaystyle 5n \:=\:35a + 10$
. . and: .$\displaystyle \dfrac{5n}{7} \:=\:\dfrac{35a+10}{7} \;=\;5a + 1 + \frac{3}{7}$
Therefore, the remainder is 3.