2005 Alabama ARML TST Problems/Problem 4

Problem

For how many ordered pairs of digits $\displaystyle (A,B)$ is $\displaystyle 2AB8$ a multiple of 12?

Solution

We wish for $2000+100A+10B+8 \equiv 0 \pmod 12\rightarrow 4A+10B\equiv 8 \pmod 12\rightarrow 2A+5B\equiv 4 \pmod 6$ . Thus $B\equiv 0 \pmod 2$ . Let $B=2C\rightarrow A+2C\equiv 2 \pmod 3$ ; $C<5$ , $A<10$ , one of the eqns. must be true: $A+2C=2\rightarrow$ 2 ways $A+2C=5\rightarrow$ 3 $A+2C=2\rightarrow$ 4 $A+2C=2\rightarrow$ 4 $A+2C=2\rightarrow$ 3 $A+2C=2\rightarrow$ 2 ways Total of 18 ways.

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2005 Alabama ARML TST Problems/Problem 4

Problem

Solution

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