1998 CEMC Gauss (Grade 7) Problems/Problem 25

Problem

Two natural numbers, $p$ and $q,$ do not end in zero. The product of any pair, $p$ and $q,$ is a power of 10 (that is, 10, 100, 1000, 10 000 , ...). If $p > q$ , the last digit of $p-q$ cannot be

$\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 9$

Solution

If the product $pq$ is a power of $10,$ and both $p$ and $q$ do not end in 0, then $p$ must be in the form $5^n$ and $q$ must be in the form $2^n.$

We know that $5^n \equiv 5 (\pmod 10)$ for all positive integers $n$ and $2^n \not\equiv 0 (\pmod 10)$ for all integers $n$ .

Therefore, we know that $p - q \not\equiv 5 - 0 = \boxed{\textbf{(C)}\ 5}.$

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1998 CEMC Gauss (Grade 7) Problems/Problem 25

Problem

Solution