506, 30602, 20306, 20603, 60203, 60302, 90805, 9090807, 8070909, &c., are multiples of 11.

Also, if we place a nought on the right of each of these numbers, as this will not affect the exactness of the division, we say that, when the sum of the figures in the even places is 11, or a multiple of 11, and there are ciphers in the odd places, the number is divisible by 11. The numbers 2090, 9020,

3080, &c., are therefore multiples of 11.

From the above it is obvious that, when a number is divisble by 11, the sum of the figures in the odd places must be equal to that of those in the even places; or the difference between these sums must be 11 or its multiple.

Also, when the number is not divisible by 11, the remainder will be, when the sum of the figures in odd places is the greater, the difference between the two sums; and, when the sum of the figures in even places is the greater, the undec. comp. of that difference.

When the first of these two sums is greater by more than 11, subtract 11 from the difference, till the remainder is less than 11. When the last is greater by more than 11, subtract the remainder from the nearest multiple of 11 that exceeds the difference.

Thus, in the number 92918, the first sum is 8+9+9 = 26; the last is 1+2=3: then 26 - 323, the difference, and 23 - 22 = 1, the remainder found in dividing 92918 by 11.

11) 92918

8447 - 1

5

Having first found the remainder, we may by it find the quotient: thus we say, 1 from 8, the right-hand figure of the dividend, 7; which we write under 8: then proceeding towards the left: 7 from 11, four, which we write under 1. Then, having added 10, we add 1 to the 4 just written, and say, from 9, four, which we write under 9. Then, 4 from 12, eight; which we write under 2. Lastly, adding 1 to 8, we say 9 from 9, nought, which we do not write. This is the countermarch of the method of multiplying by 11, (126).

In the number 909281, the sums change places, the last being 26, and the first 3. Having found the difference 23, we subtract it from 33, the nearest multiple of 11 that exceeds it thus, 33-23=10, the true remainder.

:

11) 909281

82661 - 10

Here we say, 10 from 11, one; 2 from 8, six; 6 from 12, six; 7 from 9, two; 2 from 10, eight.

178. The sum found by adding a number, consisting of an even number of figures, to the same number reversed, is ‹ multiple of 11. Thus 6718+ 8176 14894, which is a multiple of 11.

By reversing the number, the figures in odd and even places change with each other, so that though 6718÷11, gives a remainder of 8, 817611, gives a remainder of 3, the undec. comp. of 8. Hence, the sum of the numbers, which includes the sum of these remainders, is a multiple of 11.

When the number of figures is odd, the difference between a number and the same number reversed is, when not a cipher, divisible by 11.

Thus, 975-579396, a multiple of 11.

Here, in reversing the number, the figures in odd and even places do not change; consequently, in dividing each number by 11, we have the same remainder, and by subtraction, the remainder is cancelled; hence, the result is divisible by 11.

But the result (175, par. 8) is divisible by 9: therefore, when the number of figures is odd, the result is divisible by 9 and 11, that is, by 99.

Also, when the number of figures is odd, and the number odd, and odd also when reversed, the difference is even, and hence divisible by 2: that is, it is divisible by 2, 9, and 11, and consequently by 198.

When an accountant, therefore, finds in his calculation an error of 198 or its multiple, this should induce him to suspect that an odd number, consisting of an odd number of figures, and having its left-hand figure odd, has been reversed.

179. The square, cube, &c. of one prime number cannot be a multiple of another prime number, much less of the square, the cube, &c. of that other number. For, if possible, let 25, which is the square of five, be a multiple of 3. Multiply 3 and 5 together, then because 3 measures 25 and 15, it will also (172) measure 10, which is their difference. Also, because 3 divides 15 and 10, it will divide their difference 5, which is absurd.

Again, let 25 be a multiple of 7, which is greater than 5.

Find the product of 7 and 5, and because 7 divides 35 and 25, it will divide 10, their difference, or its multiple 20. Also because 7 divides 25 and 20, it will divide their difference 5, which is absurd. Now if the square, cube, &c. of a prime number cannot be a multiple of another prime number, it is evident that it cannot be a multiple of any of the powers of that other number. Wherefore, because the powers of different primes have no common factor, (that is to say, because no number can be found that will divide any two of them,) they are said to be prime to each other. Hence, the numbers 8, 9, 25, 49, &c. are prime to each other. Hence also, the squares, cubes, &c. of numbers prime to each other, are still prime to each other.

180. The product of two or more primes, can never be a multiple of any other prime number. Also, if several numbers be primes or primes to each other, no number less than their product can be their common multiple. First, if possible, let the product of the two primes 11 and 5 be a multiple of 7. Multiply 7 and 5 together. Then, because 7 divides 55 and 35, it will divide their difference 20, or its multiple 60 Also, because 7 divides 60 and 55, it will divide their difference 5, which is absurd.

Wherefore, if several numbers be prime to each other, the product of any two of them is prime to a third, and consequently, that of any three of them is prime to a fourth and so on: therefore, when several numbers are primes or primes to each other, their product is their least common multiple. Hence, it is plain that one number cannot measure another, unless all its factors are factors of that other.

SECTION VIII.

PRIME FACTORS-LEAST COMMON MULTIPLE-GREATEST COMMON MEASURE, RATIO, &c.

181. To find the prime factors of any composite number, divide the given number by the least prime number by which this can be done. Divide the quotient by the same prime, if possible. Continue to divide each successive quotient by the same prime till this is no longer possible. Then divide in the same manner, by the next least prime till it is exhausted, and then by the next, and so on, till the quotient is a prime num

461281

ber. The several divisors and last quotient will show the prime factors of the given number and the highest power of each. For example, we find the prime factors of 2520 as follows:

For practice, the prime factors of the following numbers may be found in like manner: 24, 64, 81, 512, 6561, and 8712.

But the student need not confine himself entirely to this routine; for, availing himself of the properties of numbers already known, he may, in many cases, abridge the operation. Thus, by adding the digits of 24, he will easily discover, that this number is composed of the factors 28 and 3: that 64, being 2323, is 26; that 81, being 92, is necessarily 3a, and so on.

182. It is evident, that any power of a number is a multiple of all the inferior powers of that number. Again, from what has been said (179 and 180) it is evident that a number which is a multiple of another number, must be a multiple of all the prime factors of that other number, each being taken in the highest power in which it is found factor in that number. Also, that when a number is a multiple of all the factors of another number, each taken in the highest power in which it is found factor, it is also a multiple of that other number.

From which we easily infer, that the least common multiple of several composite numbers is the product of the several prime numbers which enter into these composite numbers as factors, each being taken only once, and in the highest power in which it is found factor in any of the numbers. For example, to find the least common multiple of 16, 24, 48, 60, 81 and 128, we find by resolving these numbers into their prime factors, as in the preceeding article, that they are respectively 24; 23.3; 2+.3; 29.3.5; 34 and 27; then taking each prime in its highest power, and rejecting the others, we have 27X3 X5, or 128X81X5= 51810, the required multiple.

What is the least common multiple of the numbers 12, 18, 21, 63, 72, 114, 216, and 231 ?

183. When the same power of the same prime number is factor in each of two composite numbers, if we divide both by that prime number, and continue the division as often as possible, it is evident that the number of divisions will be the index of the power. Again, if any number of multiples, containing different powers of the same prime, be divided by that prime, and the division continued as long as the prime will divide any two of them, this division will exhaust all the inferior powers of the prime, and the quotient arising from the division of the multiple containing the highest power will still retain the prime, as factor, the number of times expressed by the difference between its index in this multiple and its index in the multiple containing its next lower power. If, therefore, this last quotient and the several divisors be multiplied together, the prime will be found factor in the product the same number of times as in the multiple containing its highest power. Therefore, to expunge, that is, to drive out or efface, all the inferior powers of the prime factors of several composite numbers, so as to find, by the multiplication of those factors, each in its highest power, the least common multiple of all the composite numbers, we have the following

Rule.

Write the numbers in a horizontal line; draw a line underneath, and divide as many as you can, which must be at least two of them, by the least prime number by which this can be done, placing those numbers which cannot be divided below the line, together with the quotients. Divide the numbers below the line in the same manner, by the same divisor, and continue so to divide as often as possible; that is, till the same divisor will no longer divide two of them. Having exhausted the first divisor, take for a divisor the next least prime that will divide two or more, with which proceed as with the first. Continue thus till the numbers below the line are prime to each other. Lastly, multiply the divisors and numbers below the line together, and the product is the least common multiple. When the quotient is a unit, it need not be written.

Example.

To find the least common multiple of 8, 9, 12, 15, 24 and 27, we place the numbers thus: