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7 trick

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7 trick

Everyone knows the 9 trick for multiplication:

  • To multiply a one digit number by 9: subtract one from the number and write down the answer in the tens (3 x 9 : 3-1 = 2 : 2x ) then subtract you new number from 9 and write down the result in the ones (9 - 2 = 7 : 27) and you have your answer.

And it's friend the "9 test" for division:

  • To test if 9 goes evenly into a number add up the digits: if the result is 9 you can divide by 9 and have no remainder (and you can get the right answer for 2 digit numbers just by adding one to the tens digit).

I have heard of a 5 test (must end in a 0 or 5) and a 3 test (digits must add up to 2 6 or 9).

But I have never heard of a 7 test. So I decided to see if I could some up with one.

Binary 7

In decimal (normal counting numbers from first grade) every digit must be 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9 (ten possible values).

In binary you get the value of a number by adding up powers of two. Every digit must be either 0 or 1 (two possible values).

In decimal (normal counting numbers from first grade) the columns are (from right to left) ones, tens, and hundreds. Just keep multiplying by 10.

234 = (4 x 1) + (3 x 10) + (2 x 100) = 4 + 30 + 200 = 234

In binary the columns are (from right to left) ones, twos and fours. Just keep multiplying by 2.

101 = (1 x 1) + (0 x 2) + (1 x 4) = 1 + 0 + 4 = 5

So in binary

111 = (1 x 1) + (1 x 2) + (1 x 4) = 1 + 2 + 4 = 7

This suggests a 7 trick:

The NEW 7 Trick

  1. To multiply a number by 7, take the original number and write it down.
    3 x 7 : 3
  2. Then double it and write down the answer under it
      3
      6
  3. Next, double the new number, and write it down
      3
      6
     12
  4. Finally add up the three numbers:
      3
      6
    +12
    ---
    +21

The reason this works is as follows:

Since

7 = 1 + 2 + 4

we can show that

N x 7 = N x ( 1 + 2 + 4 )

it follows that

N x 7 = (N x 1) + (N x 2) + (N x 4)

the steps give you the original N (3 in our example) then the original N x 2 (6 in our example) then the last number doubled: N x 2 x 2 = N x 4 (12 in our example).

The NEW "7 test" trick

  1. Divide the number by 8 (ignore the remainder for now)
    • Half the number three times: (easy for even numbers, for odd numbers just drop down one... we are ignoring the remainder)
    (first halving)  87 is odd, half of 86 is 43
    (second halving) 43 is odd, half of 42 is 21
    (third halving)  21 is odd, half of 20 is 10
  2. Jot this number down at the edge of the paper
    10
  3. Multiply the new number by 7 (using any method you know... see above for a trick)
    10 x 7 = 70
  4. Subtract the result from the original number
    87 - 70 = 17
  5. Repeat the above steps for the new number
    (first halving)  17 is odd, half of 16 is 5+3 = 8
    (second halving)  8 is even, half of 8 is 4
    (third halving)   4 is even, half of 4 is 2
  6. Jot this number down at the edge of the paper below the other intermediate result
    10
     2
  7. Multiply the new number by 7 (using any method you know... see above for a trick)
    2 x 7 = 14
  8. Subtract the result from the original number
    17 - 14 = 3
  9. Keep repeating until the result is smaller than 7
    3 is smaller than 7 so stop
  10. This left-over number is the remainder, circle it and add a letter "R"
    ( R 3 )
  11. You have completed the test... Remainder is not "0" to 87 is not evenly divisible by 7.
  12. Special Case.... Remainder is ... 7?
    • Because we are dividing by 8 instead of 7 (for a low approximation of dividing by 7) the remainder may sometimes end up as 7
    • This happens because 7 divided by 8 is 0 (8 doesn't go into 7).
    • If this happens, the remainder is really 0, and the number is evenly divisible by 7.

Going Further... division by 7

  1. Add up the numbers at the edge of the paper to get the dividend
      10
     + 2
    -----
      12
  2. You now have your answer (and not a multiplication table in sight)
    87 / 7 = 12 ( R 3 )
  3. Special Case .... Remainder is ... 7?
    • if the remainder is 7 then add one to the dividend and change the remainder to "0".
  4. Final Check
    • Use The 7 trick above to multiply the final dividend by 7 and subtract it from the original number. You should get the remainder.
      • If not ... check you math.

Optional: If you want the answer in decimal

  1. If you want the answer in decimal, like money, decide how many places you want to get (say 2 or 3) to the right of the decimal point
    I want an answer with 2 places to the right of the decimal : 12.xx (accurate to 2 places)
  2. Add this many zeros to the right of the reminder
    ( 3 R ) -> 3 with 2 zeros after it = 300
  3. Repeat the process above for this new number
    half 300 = 100+50 = 150
    half 150 = 50+ 20+5 = 50 + 25 = 75
    odd 75 -> half 74 = 30+5 + 2 = 37
  4. Save
    37
  5. Then
    37 x 7 = 37 + 74 + 148 = 111 + 148 = 259
    300 - 259 = 41
  6. Repeat
    odd 41 -> half 40 = 20
    half 20 = 10
    half 10 = 5
  7. Save
    37
     5
  8. Then
    5 x 7 = 5 + 10 + 20 = 5 + 30 = 35
    41 - 35 = 6
    6 is smaller than 7 so stop
    ( R 6 )
     37
    + 5
    ----
     42 ( R 6 )
  9. take your new number and write it with a decimal point on it's RIGHT
    42.
  10. we wanted accuracy to 2 places so wright in 2 zeros on the LEFT
    0042.
  11. Now move the decimal point 2 places to the left
    00.42
  12. Add the original dividend, place a decimal and 2 zeros on the RIGHT of it
      12.00
    + 00.42
    -------
      12.42

Optional: Rounding

  1. If you want your answer to be the closest it can be (to the true answer) with only 2 digits: you may round.
  2. If the final Remainder is 4 or more round up. If the final Remainder is 3 or less leave it alone (called rounding down).
    ( R 6 ) -> 6 is more than 4 -> round up
  3. To round up, just add one to the last digit on the right
    12.42 -> 2 + 1 = 3 -> 12.43
    87 / 7 = 12.43 (rounding to 2 places of accuracy)

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This page was last modified on 17 July 2008, at 18:26.
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