Tricks with Arithmetic

What is 26 × 34? What about 37 × 13? There are some neat little tricks you can remember which will help you do these and other calculations in your head in seconds. Here we give examples and explain why they work. The Difference of Squares Multiplying 26 by 34 might not seem to have anything to do with square numbers, but look at it like this: 26 × 34 = (30 – 4)(30 + 4) = 30² + (30 × 4) – (4 × 30) – 4² The middle two terms cancel, giving 26 × 34 = 30² – 4², and we know that 30² = 900 and that 4²= 16, so our answer is therefore 884. That was much easier than trying to do a huge multiplication! This method works in general; for two numbers a and x, (a – x)(a + x) = a² – x². So, to give another example, 17 × 23 = 20² – 3² = 391. Know Numbers My sister once said “Numbers are my friends.” She’s never been able to live it down, of course, but she may have had a point. If you get to know some interesting habits of a few numbers, mental arithmetic certainly becomes a lot simpler. For instance, it’s very useful to know that 17 × 3 = 51, and not only because it reminds you that 51 isn’t prime. For example: 24 × 17 = (8 × 3) × 17 = 8 × (3 × 17) = 8 × 51 = 408. To return to the second problem at the beginning of the article, the number 37 is interesting because 37 × 3 = 111. To write this more usefully, 37 × 3 = (100 × 1) + (10 × 1) + (1 × 1). So, 37 × 3a = a ((100 × 1) + (10 × 1) + (1 × 1)) = 100a + 10a + a . This means that 37 × 12 = 400 + 40 + 4 = 444, and so 37 × 13 = 444 + 37, which is 481. Now try 37 × 42. General points * Try factorising the numbers involved; 512 might be a lot easier to deal with if you remember it’s a power of 2. * Check your short cuts. 37 × 12 can’t be much different from 40 × 10 = 400, so if you get something wildly different, there’s been a slip at some point. * Keep practising! Here are some mental arithmetic magic tricks I have found that you can use to surprise your family. 1. if a number is divisible by 3 then so are the numbers based on mixing up the digits of the original number. For example, consider 123 which can be evenly divided by by 3. Then 132, 213, 231, 312 and 321 (which are obtained by mixing up the digits 1, 2 and 3 that make up 123) are all divisible by 3. This is called a permutation of the digits of a number. Check it for yourself! 2. to make up a number that is divisible by 4, make up a number and tag on the end any 2 digit number divisible by 4. For example, I make up the number 111111111, and now I tag 16 (which is divisible by 4) on the end to get 11111111116. This number is divisible by 4. Check it for yourself! An interesting trick follows on from this one. The following numbers can all be evenly divided by 4: 116, 1116, 11116, 111116, and so on . . . Not what you would expect! 3. if a number is divisible by 6, then any shuffle of its digits will give you a new number divisible by 6 as long as the last digit is even. For example, 1272 is divisible by 6. Permutations of its digits while keeping the last digit even gives me 2172, 2712, 1722, 7122, 7212 which can all be evenly divided by 6. Check it for yourself! 4. to make up a number that is divisible by 8, the process is very much like point 2. above. Make up a number and tag on the end any 3 digit number divisible by 8. For example, I make up the number 777777, and now I tag 016 (which is divisible by 8) on the end to get 777777016. This number is divisible by 8. Check it it for yourself! Another interesting trick follows on from this one. The following numbers can all be evenly divided by 8: 1016, 11016, 111016, and so on . . . Again, not what you would expect! 5. if a number is divisible by 9 then so are the numbers based on mixing up the digits of the original number. For example, consider 189 which is divisible by 9. Then so are 198, 819, 891, 918 and 981. Check it it for yourself! 6. if a number is divisible by 11, then permutations of its odd digits and/or its even digits will give you a new number also divisible by 11. For example, consider 154 which is divisible by 11. Then so is 451 (obtained by swapping its first and third digits). Another example, consider 1122 which is divisible by 11. Then so is 1221 (obtained by swapping its second and fourth digits). Check it it for yourself! 7. if a number is divisible by 12, then any permutations of its digits (except for the last 2) will give you new numbers also divisible by 12. For example, 14652 is divisible by 12. Then so are 16452, 41652, 46152, 61452 and 64152. Check it it for yourself! You would have to agree that such tricks do look like arithmetic magic which you can do in your head. In case you are wondering ‘why is it so?’ The trick lies in the test that determines whether a number is divisible by another.