# Short Division by 2, 3, 5, 11, 17, and 36 with and without Remainders

Ever since I was in 2nd grade and became fascinated by division, my teachers were relentless in their preoccupation with long-division. I could not believe that this was the fastest way to do division without a calculator so I endeavored to develop my own strategy I call “short division.” Why is it better? Because it is short, and not long. So simplistic.

For example, when I divide the number 987,654,321 by 2 using long division, it could take three pages of work and 5-10 minutes; when using short division, it takes under 10 seconds once you get the concept.

For the K-1st grade group, I focused on multiples of 2 that did not require remainders so they could divide 8,640,286 by 2 in a matter of seconds because they know that 2 goes into 0, 0 times, 2 goes into 2, 1 time, 2 goes into 4, 2 times, 2 goes into 6, 3 times, 2 goes into 8, 4 times. So they would get a quotient (answer to a division problem) of 4,320,143. A few of our K-1st graders I was able to challenge with remainders so they could divide 348 by 2 to get 174 using the following method.

The only difference between my method of short division and long division is that I train myself to do more of the subtraction in my head. When you ask yourself, how many times does 2 divide into 7 you immediately see 6 as the product of 2x3 and then can compare your product, 6, against the 7 to see that you have a remainder of 1. That 1 remainder is the tens number for your next division question.

It is easier to explain in steps:

STEP ONE: CREATE A TABLE OF THE FIRST NINE MULTIPLES OF THE DIVISOR, so for dividing by 4 you would write, 4,8,12,16,20,24,28,32, and 36. You will never need more than nine multiples of your divisor for this method.

If we are dividing 3,148 by 4….

STEP TWO: HOW MANY TIMES DOES 4 GO INTO 3?  THE ANSWER IS 0, SO WE PUT THE  0 IN THE QUOTIENT ABOVE THE 3 AND CARRY THE REMAINDER OF THE UNUSED 3 TO THE TENS COLUMN OF THE NEXT DIGIT IN
THE DIVIDEND (IN THIS EXAMPLE THE "1" BECOMES "31")

STEP THREE: NOW WE ASK, HOW MANY TIMES DOES 4 GO INTO 31?  LOOKING AT THE LIST OF MULTIPLES OF 4, THE NUMBER SMALLER THAN 31 INTO WHICH 4 DIVIDES IS 28 (4 GOES INTO 28, 7 TIMES. SO, WE WRITE 7 IN THE QUOTIENT ABOVE THE "1".  AFTER YOU TAKE 28 AWAY FROM 31, WHAT IS
LEFT? WE CALL THIS THE "REMAINDER."  IN THIS CASE, THE REMAINDER IS 3, THE DIFFERENCE BETWEEN 28 AND 31.  WE PUT THE 3 IN THE UPPER LEFT HAND CORNER OF THE 4 (IN THIS EXAMPLE, THE "4" BECOMES “34."

STEP FOUR: NOW WE ASK, HOW MANY TIMES DOES 4 GO INTO 34?  LOOKING AT THE LIST OF MULTIPLES OF 4, THE NUMBER SMALLER THAN
34 INTO WHICH 4 DIVIDES IS 32 (4 GOES INTO 32, 8 TIMES). SO, WE
WRITE 8 IN THE QUOTIENT ABOVE THE "4".  AFTER YOU TAKE 32 AWAY
FROM 34, WHAT IS LEFT? IN THIS CASE, THE REMAINDER IS 2, THE
DIFFERENCE BETWEEN 32 AND 34.  WE PUT THE 2 IN THE UPPER LEFT
HAND CORNER OF THE 8 (IN THIS EXAMPLE, THE "8" BECOMES "28.");
NEXT, WE ASK, HOW MANY TIMES DOES 4 GO DIVIDE INTO 28; BY NOW
WE SHOULD KNOW THAT THE ANSWER IS 7.  WE WRITE 7 ABOVE THE 8.

STEP FIVE: SINCE THERE IS NO REMAINDER WHEN WE DIVIDE 4 INTO 28, WE ARE DONE. OUR QUOTIENT (THE ANSWER) IS 787; WE CAN STATE THAT 3,148
DIVIDED BY 4 IS 787.

STEP SIX:  IN THE EVENT THAT YOU HAVE A REMAINDER AT THE END OF
YOUR CALCULATIONS, PUT THE REMAINDER AS THE NUMERATOR (TOP OF A
FRACTION) OF A FRACTION NEXT TO THE QUOTIENT WITH THE  DENOMINATOR (BOTTOM OF A FRACTION) AS THE DIVISOR.

The attached pdfs have examples at the beginning of each challenge page. We start by dividing by 2, then , 3, then 5, then for the 5th graders, dividing by 11, then 17, then 36. Remember to write down your first nine multiples of your divisor before you start.

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