*Reason it works* Take a two digit number 10n + m, where n is the tens digit and m is the units digit. The other number has to be 10n + (10 - m), since the tens digit n needs to be the same and the sum of the units digits needs to be 10. Now we multiply: (10n + m) * (10n + 10 - m) = 100n² + 100n - 10nm + 10nm + 10m - m² = 100n(n + 1) + m(10 - m). So the product of the two numbers is 100n(n + 1) + m(10 - m). Hence, the digits and tens units are occupied by m(10 - m), which is the product of the two original units digits, and the hundreds and thousands digits are the product of n by n + 1.
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*Reason it works*
Take a two digit number 10n + m, where n is the tens digit and m is the units digit.
The other number has to be 10n + (10 - m), since the tens digit n needs to be the same and the sum of the units digits needs to be 10.
Now we multiply: (10n + m) * (10n + 10 - m) = 100n² + 100n - 10nm + 10nm + 10m - m² = 100n(n + 1) + m(10 - m).
So the product of the two numbers is 100n(n + 1) + m(10 - m). Hence, the digits and tens units are occupied by m(10 - m), which is the product of the two original units digits, and the hundreds and thousands digits are the product of n by n + 1.