

Greatest Multiple in Division
Name: Elaine
Status: educator
Grade: 912
Country: Canada
Date: Summer 2013
Question:
I enjoyed a short article by Vince Calder (on your site) on fast multiplication algorithms. In the last paragraph, the author indicated that there is a similar algorithm for long division that eliminates the necessity of finding the greatest multiple in division. The author indicated that this is very "cool". I would love to read about it, but I could not find it on your site at all.
Replies:
This is a little tricky to format with the limited fonts available but here goes. Consider: 23 / 19 = 1.210526316… (irrational, so the digits keep going without a remainder or a repeating decimal). 1
.1 2
1 .1 1 02
.
19 ) 23.0 0 0
19
4 0
1 9
2 1 since 21 > 19, divide again in the same column (Remember, each place is a factor of 1/10 the previous column
19
2 0 since 2 < 10, bring down a “0” from the next column to the right
19
10 since 1 < 10 so bring down a “0” from the next column to the right, but 10 is still < 19 so the only divisor is “0”, so bring down another “0”
0
100 use a trial divisor of “2”
38
62 since 19 < 62 use another trial divisor, let’s say “2” again
38
24 since there is no divisor giving a positive remainder i.e. 19 < 24 the only trial divisor is “1”
19
5…. Now add all the trial divisors in each column. This gives 1.2105 ….
You just keep going with trial divisors in a given column, and add the columns. The remainders must be less than or equal to “0”, or repeating, never negative. But you need not use the “greatest” divisor so long as the remainder is greater or equal to “0”. Hope the formatting does not obscure the simplicity of trial divisors.
Vince Calder
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Update: November 2011

