Multiplying two-digit numbers with the same ones digit and tens digits that add up to 10
Haven't done a #MathTrickYouDidn'tKnowYouNeeded in a while, but I think I picked a good one to return with. Today, we're multiplying two two-digit numbers that share the same ones digits, and which have tens digits that add up to 10.
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As with many of these two-digit multiplication tricks, this relies on a particular polynomial expansion and some shortcuts it allows us to take. In this case, imagine writing the numbers as 10a + k and 10b + k (in which case k is the ones digit and a and b are the tens digits). When we expand out the product:
(10a + k)(10b + k)
…we get 100ab + 10k(a+b) + k². Since we know the sum of the tens digits is 10, the (a+b) can be replaced by 10, and that gives us 100ab + 100k + k². Finally, with a little clever factoring, we can write that as 100(ab+k) + k². The "100" essentially moves the ab+k term to the hundreds digit and beyond, leaving it as our product's first two digits. Since k² is the square of a digit, it can't be greater than 81, and therefore it will always take up the final two digits of the product.
#mtydkyn #mentalmath #numbersense
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