This equation contains the numbers 1-8.

Can you complete it?

6.

.. x

————

....

Puzzle Copyright © Kevin Stone

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Answer

64 x 58 = 3712.

Reasoning

Since no digit is duplicated, neither number can end in 1, otherwise, the last digit of the answer would already have been used.

Neither number can end in 5 because the answer would then end in 5 or 0.

So the first number can only be 62, 63, 64, 67, or 68.

We can now look at what the second number can end with, and we find that...

if the first number was 62, the second number can only end in 4 or 7.

Why...

not 1 as previously explained

not 2 because we've already used that in the 62

not 3 because 62 x *3 would end in 6, which is already in the 62

not 5 as previously explained

not 6 because we've already used that in the 62

not 8 because 62 x *8 would end in 6, which is already in the 62

We can repeat this for the other possible first numbers and find that...

if the first number was 62, the second number can only end in 4 or 7.

if the first number was 63, the second number can only end in 4, 7, or 8.

if the first number was 64, the second number can only end in 2, 3, 7, or 8.

if the first number was 67, the second number can only end in 2, 3, or 4.

if the first number was 68, the second number can only end in 3 or 4.

Let's check these in turn...

If the first number was 62, the only possible values are:

62 x 14 = 868

62 x 34 = 2108

62 x 54 = 3348

62 x 74 = 4588

or

62 x 17 = 1054

62 x 37 = 2294

62 x 57 = 3534

62 x 87 = 5394

Only 4 calculations are required for each option, as we didn't need to check 62 x 84 as we already know that the answer would end in 8, or 62 x 47 as we already know that the answer would end in 4.

All of these answers fail as they don't contain all of the digits.

Similarly, we can look at 63, then 64, etc. For each of the numbers, we have to check the possible endings, but in each case, only 4 calculations are required (as seen in the examples above).

We (quickly?) find 64 x 58 = 3712.

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