Algebraic Number Tricks

Test for Divisibility by Three

• You can look at a number and see if it can be divided by three. If the sum of the digits is divisible by three, so is the number. For example, 147 is divisible by three because 1 + 4 + 7 = 12, which is divisible by three. Also 512 is not divisible by three because 5 + 1 + 2 = 8, which is not divisible by three. To see why this works, consider the three-digit number ABC. This number is really 100A + 10B + C, which could also be written (99A + 9B) + (A + B + C). Clearly, 99A is divisible by three, and so is 9B, which means that (99A + 9B) is also divisible by three. Given that, ABC will only be divisible by three if (A + B + C) is also divisible by three.

Multiplication by Eleven

• Multiplying AB by 11 produces a three-digit number where the first digit is A, the second is (A + B) and the third is B. Multiplying 23 by 11 produces 253. Similarly, 34 X 11 = 374 and 11 X 11 = 121. This works because AB X 11 = (10A + B) X (10 + 1) = 100A + (10A + 10B) + B = 100A + 10(A + B) + C, and that equals the three-digit number A(A + B)C. The trick can be extended like this: 11 X ABC gives you a four-digit number where A is the first number, (A + B) is the second, (B + C) is the third and C is the fourth -- so 11 X 444 = 4884. This trick can be used when multiplying any number with repeating digits. For example 33 X 44 = 11 X 11 X (3 X 4) = 11 X 11 X 12 = 11 X 132 = 1452. This may be more steps than just multiplying 33 X 44, but all the steps are so easy you can do them in your head.

Squaring Numbers that end in Five

• To square a number that ends in five, use this schemata: For A5^2, the answer has A multiplied by (A+1) as its first digits, and 25 as the remaining digits, so 35^2 will be (3 X 4) joined with 25, which is 1225. Similarly, 75^2 = 5625. This works because A5^2 = A5 X A5 = (10A + 5)(10A + 5) = 100A^2 + (50A + 50A) + 25 = 100A^2 + 100A + 25 = 100(A^2 + A) + 25. This idea can be extended to larger numbers, so 115 ^2 = (11 X 12) joined with 25 = 13225.

Multiplying Numbers that are Two Different

• When numbers differ by two, and the number between them is easy to square, you can use the algebraic identity Z^2 - 1 = (Z -1)(Z + 1) to simplify the multiplication. For example, 14 X 16 = 15^2 - 1 = 225 - 1 = 224. Similarly, 34 X 36 = 35^2 - 1 = 1225 - 1 = 1224. When lightning-fast calculators multiply large numbers instantly, they are not working faster -- they are doing less work.