Homework help 1 – divisibility rules (elementary)

(The first in a semi-regular series. A math boot camp to help you help your kiddo, if you will.)

So. Your grade-schooler brought home a homework sheet that asks questions like: “What are the factors of blankity-blank?” and “What are the multiples of mumble-mumble?” with the word prime thrown in there somewhere.

First: some vocab.

  • Divisors: numbers that go into what you are given without a remainder. Example – divisors of 8 are 1, 2, 4, and 8
  • Multiples: numbers that you get by multiplying what you are given over and over. (Think of multiples as one number’s list on the times table). Example – multiples of 8 are 8, 16, 24, 32…
  • Divisors make a list that ends. Multiples go on forever and ever.
  • Factors is a fancy-schmancy word for divisors

When the question asks “What are the factors of 100?”, they are asking you to list any and all numbers that go into 100 without a remainder. I do this by making times-table pairs, and then writing them all in order. 100 = 1 x 100 or 2 x 50 or 4 x 25 or 5 x 20 or 10×10 and that’s all. So the answer to “What are the factors of 100?” is 1, 2, 4, 5, 10, 20, 25, 50, 100.

Sometimes they try to trick you: “What are the factors of 17?” Nothing goes into 17, you say (smarty-pants). Correct – but any number can be divided by 1 and by itself. So to answer “What are the factors of 17?” you list 1, 17.

The hard part comes when the numbers are big. Listing the pairs for 100 was intuitive. Listing the pairs for 1350? Not so much. This is where a few tricks come in.

  1. Figure out an approximate square root for the number you’re given, rounding up. If I’m looking at 1350, I can use the calculator to find that square root of 1350 = 36.7ish. If I didn’t have a calculator, I would say that 40×40 is 1600, which is close enough for our purposes.Why? Because each pair of numbers (like 1×100, 2×50, etc in our earlier example) has a small and a big number in it. The small number MUST be less than or equal to the square root. So for 1350, I don’t need to check any numbers bigger than 37. This is why a close guess that’s a little big (like 40) is fine too. I’m just trying to cut down on the number of things I’ve got to check.
  2. So we know with 1350 that we don’t have to check any numbers bigger than 37. There are some tricks for checking some of the other numbers, without punching anything into a calculator:
    • 2: is the number even (ending in a 0,2, 4, 6 or 8)? If so, 2 is a divisor. If not – 2 isn’t, and NEITHER IS ANY OTHER EVEN NUMBER. This is a big time-saver. Again: if 2 doesn’t go into a number, neither can 4, 6, 8, 10, etc. In our problem 1350 ends in 0, so the number IS divisible by 2. 2 x 675 = 1350
    • 3: if you add up all the digits, do you get a 3, 6, or 9? In our problem (1350), 1 + 3 + 5 + 0 = 9. So yes, 1350 IS divisible by 3. 3 x 450 = 1350
    • 4: look at the last two digits of your number. Is this 2-digit number on the 4 times table? In our problem (1350), 50 is NOT on the 4’s times table. So 1350 is NOT divisible by 4. (Remember, if you’d answered NO to 2, then you could completely skip checking 4).
    • 5: does the number end in a 5 or a 0? In our problem (1350), yes. So 1350 IS divisible by 5. 5 x 270 = 1350
    • 6: Was the number divisible by BOTH 2 & 3? (One or the other is not good enough). Our problem (1350) WAS divisible by 2 & 3, so it is automatically divisible by 6. 6 x 225 = 1350
    • 7: The tests for 7 suck. More work than they are worth – it’s easier to just use long division. Does 7 divide into 1350 with no remainder (or no decimal on the calculator)? In our case (1350), no. So 1350 is not divisible by 7.
    • 8: Do the last 3 digits make a number that is on the 8’s times table? This rule still requires long division, but a little bitty problem instead of a big one. In our problem (1350), 350 divided by 8 gives us a decimal. So 1350 is NOT divisible by 8.
    • 9: Do the digits add up to 9? (This is JUST like the 3’s test, but instead of good options of 3, 6, & 9, the only way to say yes is if they add up to 9.) For ours (1350), we say earlier that 1 + 3 + 5 + 0 = 0, so yes 1350 IS divisible by 9. 9 x 150 = 1350
    • 10: Does the number end in a 0? (Again, just a slightly more-restrictive version of the 5’s test.) 1350 ends in a 0, so yes, 1350 IS divisible by 10. 10 x 135 = 1350

    So far we have pairs of 1 x 1350, 2 x 675, 3 x 450, 5 x 270, 6 x 225, 9 x 150 and 10 x 135. We need to check the numbers from 11 to 37 by hand. Using my calculator, I get that 15 x 90, 18 x 75, 25 x 54, 27 x 50 and 30 x 45. The only other speed-up trick is that if a number is not a divisor, neither are its multiples. In 1350, I know that 7 is not a divisor. That means I don’t have to check 14, 21, 28 or 35.

    So the answer to the question “What are the divisors of 1350?” is 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 27, 30, 45, 50, 54, 75, 90, 135, 150, 225, 270, 450, 675, 1350.

Bonus example

  • Find all the divisors of 225
    first: 15 x 15 = 225 (aka, the square root of 225 is 15). Thus I don’t need to check anything bigger than 15.
    2? Nope, not even. This also rules out 4, 6, 8, 10, 12 and 14
    3? 2 + 2 + 5 = 9, so yup. 3 x 75 = 225
    4? skip
    5? ends in a 5. Yup. 5 x 45 = 225
    6? skip
    7? by long division, 7 goes into 225 32 times with a remainder of 1. Nope.
    8? skip
    9? 2 + 2 + 5 = 9, so yup. 9 x 25 = 225
    10? skip
    11? by long division, 11 goes into 225 20 times with a remainder of 5. Nope.
    12? skip
    13? by long division, 13 goes into 225 17 times with a remainder of 4. Nope.
    14? skip
    15? by long division, 15 goes into 225 15 times with a remainder of 0. Yup. 15 x 15 = 225.
    list of divisors of 225: 1, 3, 5, 9, 15, 225, 45, 75, 225

Next time: PRIME factors (:

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2 Comments on “Homework help 1 – divisibility rules (elementary)”

  1. Laurie says:

    Sure could have used this in 3rd-5th grades when helping–old “guess and check” is a bit tedious….


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