DIVISIBILITY OF NUMBERS 



1816 



DIVISIBILITY OF NUMBERS 



divisible by 11, the number is divisible by 11; 

 for example, 819192. (8+9+9) (1+1+2) =22. 

 22 is divisible by 11 ; 819192 is divisible by 11. 

 819192+11=74472. Another illustration: 455488. 

 The difference between the sums of the alter- 

 aate digits is (4+5+8=17) (5+4+8=17) =0, 

 and the number is divisible by 11. 455488 -HI 

 =41408. 



Divisibility by 7. The test is so much more 

 difficult than the division itself is, except for 

 very long numbers, that the test is used rarely. 

 It is given below: 



Take the last six digits at the right as a 

 number; take the next six digits as a number 

 and so on. Find the sum of these numbers. 

 If the sum is divisible by 7, the number is 

 divisible by 7; for example, 344519,485268. 



71829787 

 118541 



Since this sum is divisible by 7, we know 

 that the number is. It is seen that for num- 

 bers of less than 7 digits, the test has no ad- 

 vantage, and has little, if any, for numbers of 

 more digits. 



Casting out of g's. This is a test used as a 

 check on work in the four fundamental proc- 

 esses. It is open to criticism as a check because 

 it will not show up an error of 9 or any mul- 

 tiple of 9. 



Test for Addition. Cast out the 9's from 

 each column, and note the remainders or the 

 excess (as it is. called) ; then cast out 9's from 

 the sum of these excesses. Then cast out, 9's 

 from the sum of all the addends. If the ex- 

 cesses obtained in both cases agree, the sum is 

 correct. For example: 



694 

 832 

 796 



458 

 2780 



4+2+6+8=2X9+2 



9+3+9+5=2X9+8 



6+8+7+4=2x9+7 

 Excesses are 2, 8, 7. 2+8+7=9+8. 8 is 

 the excess after casting 9's from the columns. 

 2+7+8+0=9+8. 8 is the excess after casting 

 9's from the sum. The excesses agree. We 

 may cast out 9's from the addends horizontally 

 instead of vertically, as follows: 



6+9+4=2X9+1 



8+3+2=1X9+4 



7+9+6=2X9+4 



; t 4 + 5 + 8 = 1X9 + 8 



Sum of excesses= 1+4+4+8 =9+8. The ex- 

 cess is 8, and agrees with the excess of the sum 

 as seen above. 



Test for Subtraction. Cast out the 9's in the 

 subtrahend, and the 9's in the difference. Add 

 the excesses found thus and cast the 9's from 

 the sum. Then cast 9's from the minuend, and 

 find the excess. If this latter excess agrees 

 with the former the answer is correct. This 

 follows from the method in proving addition, 

 since the minuend is the sum of the subtra- 

 hend and difference. 



Test for Multiplication. Cast out 9's in the 

 multiplicand and find the excess; cast out 9's 

 in the multiplier and find the excess; find the 

 product of these excesses and cast out 9's from 

 it, and note the excess. Now cast out 9's from 

 the product and note the excess. If these last 

 excesses agree, the product is correct; for ex- 

 ample : 



98743 

 6582 



197486 

 789944 

 493715 

 692458 

 649926426 



=3X9 + 4 



(6 + 5 + 8 + 2)=2X9 + 3 

 4X3 = 12 = 9 + 3 



Excess is 3. (6+4+9+9+2+6+4+2+6) =5 

 X9+3. Excess is 3. Excesses agree; product 

 is correct. We may also test the addition of 

 the partial products as one would test any ad- 

 dition. 



Test for Division. Cast out 9's in divisor and 

 note the excess; cast out 9's in the quotient 

 and note the excess. Find product of these 

 excesses; cast out 9's and note excess. Then 

 cast out 9's from the dividend and note the 

 excess. If these last two excesses agree the 

 quotient is correct. In case of a division 

 which gives a remainder, cast out 9's from the 

 remainder and note the excess obtained. Add 

 this to the excess obtained after casting 9's 

 from the product of the divisor and quotient, 

 and this sum must agree with the excess in the 

 dividend. Problem: Make use of test in prov- 

 ing correctness of quotient in example, 74947352 

 +974=76948. 



Illustration of first case : 



76948 



974|74947352 

 6818 

 6767 

 5844 



9+7+4=2x9+2 



7+6+9+4+8=3X9+7 



2x7=9+5 



9233 

 8766 

 4675 

 3896 

 7792 

 7792 



Excess is 5. 7+4+9+4+7+3+5+2=4X9+5, 

 Excess is 5. Quotient is correct. 



