2005 Alabama ARML TST Problems/Problem 4

Problem

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

Solution

We wish for $2000+100A+10B+8 \equiv 0 \pmod {12}\Longleftrightarrow 4A+10B$ $\equiv 8 \pmod {12} \Longleftrightarrow 2A+5B\equiv 4 \pmod 6$ . 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 ways

$A+2C=2\rightarrow$ 4 ways

$A+2C=2\rightarrow$ 3 ways

$A+2C=2\rightarrow$ 2 ways

We have a total of $18$ ways.

2005 Alabama ARML TST (Problems)
Preceded by: Problem 3	Followed by: Problem 5
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2005 Alabama ARML TST Problems/Problem 4

Problem

Solution

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