By Paul Ramirez
In this excerpt from Mental Math, the authors review divisibility rules and discuss the Rule of 70 and Rule of 110:
Testing for Divisiblity:
We end this chapter with a brief discussion of how to determine whether one number is a factor of another number. Being able to find the factors of a number helps us simplify division problems and can speed up many multiplication problems. This will also be a very useful tool when we get to advanced multiplication, as you will often be looking for ways to factor a two–, three–, or even a five–digit number in the middle of a multiplication problem. Being able to factor these numbers quickly is very handy. And besides, I think some of the rules are just beautiful.
It’s easy to test whether a number is divisible by 2. All you need to do is to check if the last digit is even. If the last digit is 2, 4, 6, 8, or 0, the entire number is divisible by 2.
To test whether a number is divisible by 4, check if the two–digit number at the end is divisible by 4. The number 57,852 is a multiple of 4 because 52 = 13 × 4. The number 69,346 is not a multiple of 4 because 46 is not a multiple of 4. The reason this works is because 4 divides evenly into 100 and thus into any multiple of 100. Thus, since 4 divides evenly into 57,800, and 4 divides into 52, we know that 4 divides evenly into their sum, 57,852.
Likewise, since 8 divides into 1000, to test for divisibility by 8, check the last three digits of the number. For the number 14,918, divide 8 into 918. Since this leaves you with a remainder (918 ÷ 8 = 114), the number is not divisible by 8. You could also have observed this by noticing that 18 (the last two digits of 14,918) is not divisible by 4, and since 14,918 is not divisible by 4, it can’t be divisible by 8 either.
When it comes to divisibility by 3, here’s a cool rule that’s easy to remember: A number is divisible by 3 if and only if the sum of its digits are divisible by 3— no matter how many digits are in the number. To test whether 57,852 is divisible by 3, simply add 5 + 7 + 8 + 5 + 2 = 27. Since 27 is a multiple of 3, we know 57,852 is a multiple of 3. The same amazing rule holds true for divisibility by 9. A number is divisible by 9 if and only if its digits sum to a multiple of 9. Hence, 57,852 is a multiple of 9, whereas 31,416, which sums to 15, is not. The reason this works is based on the fact that the numbers 1, 10, 100, 1,000, 10,000, and so on, are all 1 greater than a multiple of 9.
A number is divisible by 6 if and only if it is even and divisible by 3, so it is easy to test divisibility by 6.
To establish whether a number is divisible by 5 is even easier. Any number, no matter how large, is a multiple of 5 if and only if it ends in 5 or 0.
Establishing divisibility by 11 is almost as easy as determining divisibility by 3 or 9. A number is divisible by 11 if and only if you arrive at either 0 or a multiple of 11 when you alternately subtract and add the digits of a number. For instance, 73,958 is not divisible by 11 since 7 − 3 + 9 − 5 + 8 = 16. However, the numbers 8,492 and 73,194 are multiples of 11, since 8 − 4 + 9 − 2 = 11 and 7 − 3 + 1 − 9 + 4 = 0. The reason this works is based, like the rule for 3s and 9s, on the fact that the numbers 1, 100, 10,000, and 1,000,000 are 1 more than a multiple of 11, whereas the numbers 10, 1,000, 100,000, and so on are 1 less than a multiple of 11.
Testing for divisibility by 7 is a bit trickier. If you add or subtract a number that is a multiple of 7 to the number you are testing, and the resulting number is a multiple of 7, then the test is positive. I always choose to add or subtract a multiple of 7 so that the resulting sum or difference ends in 0. For example, to test the number 5292, I subtract 42 (a multiple of 7) to obtain 5250. Next, I get rid of the 0 at the end (since dividing by ten does not affect divisibility by seven), leaving me with 525. Then I repeat the process by adding 35 (a multiple of 7), which gives me 560. When I delete the 0, I’m left with 56, which I know to be a multiple of 7. Therefore, the original number 5292 is divisible by 7.
This method works not only for 7s, but also for any odd number that doesn’t end in 5. For example, to test whether 8792 is divisible by 13, subtract 4 × 13 = 52 from 8792, to arrive at 8740. Dropping the 0 results in 874. Then add 2 × 13 = 26 to arrive at 900. Dropping the two 0s leaves you with 9, which is clearly not a multiple of 13. Therefore, 8792 is not a multiple of 13.
Some “Interest–ing” Calculations
Finally, we’ll briefly mention some practical problems pertaining to interest, from the standpoint of watching your investments grow, and paying off money that you owe.
We begin with the famous Rule of 70, which tells you approximately how long it takes your money to double: To find the number of years that it will take for your money to double, divide the number 70 by the rate of interest.
Suppose that you find an investment that promises to pay you 5% interest per year. Since 70 ÷ 5 = 14, then it will take about 14 years for your money to double. For example, if you invested $ 1000 in a savings account that paid that interest, then after 14 years, it will have $ 1000( 1.05) 14 = $ 1979.93. With an interest rate of 7%, the Rule of 70 indicates that it will take about 10 years for your money to double. Indeed, if you invest $ 1000 at that annual interest rate, you will have after ten years $ 1000( 1.07) 10 = $ 1967.15. At a rate of 2%, the Rule of 70 says that it should take about 35 years to double, as shown below:
$1000( 1.02)35 = $1999.88
A similar method is called the Rule of 110, which indicates how long it takes for your money to triple. For example, at 5% interest, since 110 ÷ 5 = 22, it should take about 22 years to turn $1000 into $3000. This is verified by the calculation $1000( 1.05)22 = $2925.26. The Rule of 70 and the Rule of 110 are based on properties of the number e = 2.71828 … and “natural logarithms” (studied in precalculus), but fortunately we do not need to utilize this.
From Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks by Arthur Benjamin and Michael Shermer, copyright © 2006 by Arthur Benjamin and Michael Shermer. Used by permission of Three Rivers Press, a division of Random House, Inc. Any third party use of this material, outside of this publication, is prohibited. Interested parties must apply directly to Random House, Inc. for permission.
Paul Ramirez, FSA, MAAA, is a senior actuarial associate at Allstate Benefits. He can be reached at firstname.lastname@example.org