FUN WITH FIGURES.

New Zealand Graphic, Volume XVI, Issue XII, 21 March 1896, Page 335

FUN WITH FIGURES.

AN odd number. Every other number is an odd number, of course, but one of the oddest of all the odd numbers is 7. I could not detail all its peculiarities in the space that the editor would permit me to occupy ; but I can call attention to one, at least, which is worthy the notice not only of boys and girls who like to deal with the curiosities of numbers, but of mathematicians also. Take any two-figured multiple of 7 —that is, any number of two figures that is produced by the multiplication of 7 by another number—say 14, 42. 98. 35, 63, 56 ; multiply the left-hand number by 5, the right-hand by 4 ; add the products, and the sum is another multiple of 7. Thus, take 21 (a multiple of 7) : 2 x 5 = 10 4xl= 4 14 2 times 7. Take 98, a multiple : 9 x 5 = 45 4 x 8 = 32 77 = 11 times 7. Take 35, a multiple : 3 x 5 = 15 5 x 4 = 20 35 5 times 7. In the case of a multiple of three figures, as 105, 413, 315, 203, etc., multiply the two left-hand figures by 5, and add 4 times the right-hand figure. The sum of the products is a multiple of 7. Take 833, which is a multiple of 7: 83 x 5 = 415 3x4= 12 427 = 61 times 7. Take 525, which is a multiple : 52 x 5 = 260 sx4= 20 280 = 40 times 7. Now here is something that you may use as an arithmetical puzzle, while at the same time it may prompt you to look into the properties of numbers—a subject that has been wofully neglected, not only by our teachers, but by those who make arithmetics for our teachers as well as for us youngsters. Let your companion write down a string of figures—as many as he pleases—and read them to you slowly. Suppose you have asked for a string of five, and suppose he reads ‘ 10,039.’ You follow the reading, and as it proceeds you divide by 7, paying no heed to the quotient, but watching the remainders, so as to find whether the number is divisible by 7. Let me explain : He reads ‘l,o;’ this to you means 10. You silently divide 10 by 7. The remainder, 3, you hold until he reads the next figure, o. This, with your 3, is 30, which is 4 times 7 with 2to spare. The 2 you hold, and he reads the next figure, 3, which with your 2 is 23. Divided by 7, it gives 3 and a remainder of 2. You hold this till he reads 9. Here you have 29 to deal with. This is 4 times 7, with a remainder of 1. Now, to make 10,039 an exact mutliple of 7, all you have to do is to subtract this remainder. Ask your companion to do this. Next say to him, ‘ Cast out the last figure, multiply the rest of the number by 3, and add the number cast out.’ This he does in silence. Do the same with the result, casting out the unit figure, multiplying the rest by 3, and adding the rejected unit. Let him continue this process as long as he can, ‘and,’ you tell him, ‘ your last result will be 7.’ Suppose we look at the work as it will stand : 1003.8 = the number, with the unit out. 301.7 the number x by 3, + 8, unitout. 91.0 the number x by 3, +7, unit out: and so on to the end. 27.3 8.4 2.8 1.4 7 If the figures read to you form a number that is divisible by 7 without a remainder you need make no change. If the number be too great, make it smaller by the difference between it and the nearest multiple. If the figures given are 1,2, 3, 4. you find 1234 too great by 2, so you cause your companion to subtract 2 as the first step in the game. If the number be 43504, you find it too small by 1; so you tell him to add 1. But, remember, the number must be a multiple of 7, and it behoves you to practice short division in order to identify it as quickly as possible. Let your mind and not your fingers do the figuring. Your teacher may be able to help you discover the why and wherefore of this peculiarity of 7. I should like to be able to explain it myself. Now as you always know the result of your companion's figuring—if it is correctly done—you can vary your prophecy in various ways, and puzzle even the great mathematicians. For example, after directing your companion as to casting out, multiplying, adding in, etc. you may say, ‘ When you come to your last figure—whatever it may be—multiply it by the product of 3, 11, 13, and 37, and the result will be a number of six figures—all alike.’ Or, ‘ Square any number that contains your last figure as a unit, subtract I, and the remainder will be a multiple of B.’