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In the multiplication below, some of the digits have been replaced by letters and others by asterisks. Where a digit has been replaced by a letter, the same letter is used each time, and different letters have replaced different digits. Can you reconstruct the original multiplication?

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The following splendid solution, from Prav at the North London Collegiate School Puzzle Club, gives a clear and full explanation of all the reasoning required to solve this problem. Very well done Prav!

The answer is as follows:

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I will now try to explain how I found my answer. We know that when ABC is multiplied by A the answer is a three-digit number. This means that A is an integer of value less than or equal to three. We also know that when A is multiplied by C the resulting answer is a number which ends in A, i.e. A * C = XA.

This means that A cannot be 1 because the only digit multiplied by 1 to end in a number whose last digit is 1 is 1 itself and A and C cannot be the same number. Also the value of C cannot be 1 because ABC multiplied by C would then be a 3 digit number instead of a 4 digit number.

We know A cannot be three because there is no digit C, which multiplied by 3, will equal X3.

Therefore our only possibility is that A is equal to two.

Because A * C = XA we can substitute A=2 into the formula. so 2 * C = X2.

The only digit which multiplied by 2 has 2 as the units digit 6, this is because 2 * 6 = 12.

We also know that B * C = XB. We now know that C = 6 so we can substitute this into the formula too.

B * C = XB

B * 6 = XB

The possibilities for B are 4, 6 and 8.

We know that B can not be 6 because C is 6 and both numbers can not be the same. Also B can not be 4 because when ABC is multiplied by B the answer is a 4-digit number but if B were 4 then we would have 246 * 4. This results in 904, which is not a 4-digit number, so the only possible remaining digit is 8. When substituted in the formula it works, as shown above.

So our final answer is that A = 2, B = 8 and C = 6!!!