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If you ignore multiplication by 10 and powers of ten, it reduces the problem to single digit multiplications, followed by addition.

For example,

38 * 47 = (30 + 8)*(40 + 7) = 30*40 + 30*7 + 8*40 + 8*7

= 3*4*100 + 3*7*10 + 8*4*10 + 8*7

where each multiplication is essentially of two single-digit numbers.

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