Fast Arithmetic Tips

Checking the result fast

  1. Divisibility by 5.

    A number is divisible by 5 iff it ends with either 5 or 0. So you know right away that 123·345=42434 is wrong. It's a different question how to get the right answer (42435).

  2. Use divisibility criteria.

    As above, divisibility criteria are useful in establishing whether a result is wrong. Note that this way you never may be sure whether the result is right. The following criteria is available:

    1. Divisibility by 3
    2. Divisibility by 9
    3. Divisibility by 7
    4. Divisibility by 11
    5. A number is divisible by 4 iff its 2-digit tail is divisible by 4. For example, 12345 is not divisible by 4 since 45 is not. However, 12348 is divisible by 4 since 48 is.
    6. A number is divisible by 8 iff its 3-digit tail is divisible by 8. For example, 12345 is not divisible by 8 since 345 is not. However, 12344 is divisible by 8 since 344 is. Indeed 344 = 320 + 24, where both addends are divisible by 8.
    7. A number is divisible by 25 if it ends with one of the four: 00, 25, 50, 75. Therefore 234465 is not divisible by 25.
    8. The is a general way of deriving divisibility criteria for numbers such as 7, 11, 13, and other.

  3. Use rude bounds.

    1. 127·345 = 27455 is clearly wrong because 127 > 100 while 345 > 300. Therefore, it must be that 127·345 > 100·300 = 30000.
    2. 220 = 524288 is clearly wrong because 220 = (210)² = 1024² > 1000² = 1000000. Actually 524288 = 219.
    3. A practical application.

  4. Use digital roots.

    Digital root of an integer is, by definition, the sum of its digits computed recursively until only one digit remains. Thus to compute the digital root of 33572, we calculate first 3 + 3 + 5 + 7 + 2 = 20. The digitl root of 33572 is then 2 = 2 + 0.

    The important fact about digital roots is that the digital root of the product of integers equals the product of their digital roots (downsized to a digital root if necessary.)

    348·11 = 3828. The digital roots of 348, 11, and 3828 are respectively 6, 2, and 3. Take the product of 6 and 2: 6·2 = 12 with the digital root equal to 1 + 2 = 3, which is exactly the digital root of 3828.

    (Thanks are due to Jayesh Patel from Gujarat State (India) for reminding me of this useful property of the digital roots.)

    The reason for this is that a digital root of a number is simply the remainder of the division of the number by 9. The operation of omputing digital roots is equivalent to the process known as "casting out 9s". Indeed, for any n ≥ 0;, 10n ≡1 (mod 9).

    Related material
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  5. A Broken Calculator Has Its Uses
  6. Fast Arithmetic Tips: Getting the Result Fast
  7. Stunning Friends With Math Magic
  8. Divisibility Criteria
  9. Abacus and Its Relatives

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