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, $10,100,1000,10000, \ldots$ ). If $p>q$, the last digit of $p-q$ cannot be
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9
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, $10,100,1000,10000, \ldots$ ). If $p>q$, the last digit of $p-q$ cannot be
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

Solution

If the two natural numbers $p$ and $q$ do not end in zero themselves and if their product is a power of 10 then $p$ must be of the form $5^{n}$ and $q$ must be of the form $2^{n}$.

This is true because $10=2 \times 5$ and $10^{n}=(2 \times 5)^{n}=2^{n} \times 5^{n}$.

The possibilities for powers of two are $2,4,8,16,32, \ldots$ and for corresponding powers of five are 5 , $25,125,625,3125, \ldots$.

If we take their differences and look at the last digit of $p-q$ we find the following,

$\left.\begin{array}{rrr}\multicolumn{1}{c}{p} & \underline{q} & \text { last digit of } p-q \\ 5 & 2 & 3 \\ 25 & 4 & 1 \\ 125 & 8 & 7 \\ 625 & 16 & 9\end{array}\right\} \quad$ and the pattern continues in groups of 4.

Thus, the last digit of $p-q$ cannot be 5 .

ANSWER: (C)