231
The number 231 has some interesting properties. First, there are 231 cubic inches in a gallon. Since 231 = 3 X 7 X 11, if you build a box with inside measurements of 3 inches, 7 inches and 11 inches, it will hold one gallon. This is about the size of an extra-large, family-size box of breakfast cereal.
231 can also be used to see if a three-digit number is divisible by 7. Take any three-digit number, abc. Multiply the first digit by 2, the second by 3 and the third by 1. Add the results; if the sum is divisible by 7, so is the original number.
As an example, take 413.
4 X 2 + 1 X 3 + 3 X 1 = 8 + 3 + 3 = 14.
Since 14 is divisible by 7, so is 413. In fact, 413 = 7 X 59.
The reason this works can be shown by examining any three-digit number, call it abc.
abc = a X 100 + b X 10 + c X 1
= a X (98 + 2) + b X (7 + 3) + c X 1
= (a X 98) + (a X 2) + (b X 7) + (b X 3) + (c X 1) , distributing the terms a and b
=(a X 98) + (b X 7) + (a X 2) + (b X 3) + (c X 1), grouping the terms
=(a X 7 x 14) + (b X 7) + (a X 2) + (b X 3) + (c X 1), factoring 98 in the first term
=7 X {(a X 14) + (b X 1)} + {(a X 2) + (b X 3) + (c X 1)}, extracting 7 from the first two terms and grouping
Since the first term, 7 X {(a X 14) + (b X 1)} is divisible by 7, having a factor of 7, the whole expression can only be divisible by 7 if the second set of terms, (a X 2) + (b X 3) + (c X 1), is divisible by 7. But the second term is the expansion of the original three digits by the digits of 231. QED
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