Problem 32

Pandigital Products

We shall say that an $n$ -digit number is pandigital if it makes use of all the digits $1$ to $n$ exactly once; for example, the $5$ -digit number, $15234$ , is $1$ through $5$ pandigital.

The product $7254$ is unusual, as the identity, $39 \times 186 = 7254$ , containing multiplicand, multiplier, and product is $1$ through $9$ pandigital.

Find the sum of all products whose multiplicand/multiplier/product identity can be written as a $1$ through $9$ pandigital.

HINT: Some products can be obtained in more than one way so be sure to only include it once in your sum.

全数字的乘积

如果一个 $n$ 位数包含了 $1$ 至 $n$ 的所有数字恰好一次，我们称它为全数字的；例如，五位数 $15234$ 就是 $1$ 至 $5$ 全数字的。

$7254$ 是一个特殊的乘积，因为在等式 $39 \times 186 = 7254$ 中，被乘数、乘数和乘积恰好是 $1$ 至 $9$ 全数字的。

找出所有被乘数、乘数和乘积恰好是 $1$ 至 $9$ 全数字的乘法等式，并求出这些等式中乘积的和。

注意：有些乘积可能从多个乘法等式中得到，但在求和的时候只计算一次。