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jasen

alphametic with modulo

Question

Solve this alphametic (replacing letters with digits)

ABCDEFG mod (A×B×C×D×E×F×G)=0

example :

ABCDE mod (A×B×C×D×E)=0
13248 mod (1×3×2×4×8)=0

Note :

  • Do not replace any letter with 0.
  • Each letter represent different digit.
  • There is only one solution.

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ABCDEFG is a multiple of the product of its digits, so it must be divisible by each of its digits. With at least two even digits, it is an even number. It is not divisible by 10, and is therefore not divisible by 5. At least two digits are multiples of 3, so the number and the sum of its digits are multiples of 9. The sum must be 36, and the digits are 1, 2, 3, 6, 7, 8 and 9.

The product is 18,144 and the number is 1,687,392.

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