ma + th = math?

Number Theory Level 3

How many ordered quadruples of positive integer solutions are there to the equation $MA + TH = MATH \, ?$

Clarification:

The above is 1 equation in 4 variables: $(M\times A)+(T\times H)=M\times A\times T\times H.$

0 1 2 3 4 5 6 7

1 solution

Brian Charlesworth
Mar 15, 2017

The given equation can be rewritten as $math - ma - th = 0 \Longrightarrow (ma - 1)(th - 1) = 1$ .

As $m,a,t,h$ are all positive we know that $ma, th \ge 1$ , so we must have

$ma - 1 = th - 1 = 1 \Longrightarrow ma = th = 2$ .

So there are two options for each of $(m,a)$ and $(t,h)$ , namely $(1,2)$ and $(2,1)$ , so for $(m,a,t,h)$ there are a total of $2 \times 2 = \boxed{4}$ options.

ma + th = math?

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