1999 CEMC Gauss (Grade 8) Problems/Problem 4

Problem

What is the remainder when $82~460$ is divided by $8$ ?

$\text{ (A) }\ 0 \qquad\text{ (B) }\ 5 \qquad\text{ (C) }\ 4 \qquad\text{ (D) }\ 7 \qquad\text{ (E) }\ 2$

We can do long division to find the answer. $8 \times 10~000 = 80~000$ , so there is $2460$ remaining.

$8$ does not go into $2$ , so we have to bring the next digit.

$8$ goes into $24$ exactly $3$ times with no remainder, so $60$ is remaining.

$8$ goes into $60$ $7$ times, but there is a remainder of $60 - 8 \times 7 = 60 - 56 = \boxed {\textbf {(C) } 4}$ .

~anabel.disher

Because $1000 = 2^3 \times 5^3$ divided by $8 = 2^3$ gives a whole number, only the last three digits matter.

$400 = 8 \times 50$ , so we have $60$ remaining. We can then see the remainder is $\boxed {\textbf {(C) } 4}$ because $60 = 8 \times 7 + 4$ .

~anabel.disher