it is the greatest number that will divide both. It is also plain that, if any number will divide both, D is that number, or a multiple of that number. Wherefore, the difference between two numbers is the greatest number that can divide both without a remainder: and, if any number will divide each of two numbers without remainder, their difference is that number or its multiple.

174. A number which cannot, without remainder, be divided by any other number, is called a prime number, or simply a prime.

In the regular formation of numbers, such numbers must arise; for, as one number cannot contain another greater than itself, it is plain that 2 is the first prime number, and that the number 3, which is odd, and, consequently, not divisible by 2, is also a prime number.

Numbers which are not prime are called composite numbers; because each is composed by the multiplication or involution of two or more primes. Wherefore, because they can be divided by 2, all even numbers except 2 are composite and terminate with 0, 2, 4, 6, or 8. Hence, every prime number, except 2, is odd; and, when it consists of more than one figure, terminates with 1, 3, 7 or 9.

The square of any prime number, except 2 and 5, terminates with 1 or 9.

CUBES.

4

8

9

27

25

125

49

343

11

121

13

169

2197

17

289

4913

so do all its

148877

powers.

When a prime ends with 1,

When it ends with 9, the

square ends with 1.

The primes which succeed 97, are 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 287, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 613, 617, 619, 631, 641, 613, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 779, 787, 791, 793, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, &c.

To determine whether a number ending with 1, 3, 7, or 9 is or is not prime, divide the number by each of the prime numbers 3, 7, 11, 13, 17, &c., till the quotient is not greater than the divisor. When every division within this limit gives a remainder, the given number is prime: because any greater divisor would give a less quotient, and this would fall within the range of numbers already tried. Thus the scholar will find that 1067 is a multiple of 11 and 97: also that 1069 is a prime number.

Suppose we would examine 14363. Having tried the numbers 3, 7, 11, 13, 17, 19, without success, we say that, as the number ends with 3, it is probably a multiple of some one of the higher primes ending with 3. We, therefore, try 23, 43, &c., thus:

By the remainder at the end of each of the two firf these operations, we see that neither 23 nor 43 is a mea re of 14363. But, as the third operation gives no remainder, we find that 53 is a measure.

Numbers which have no common factor or measure are said

to be prime to each other. Thus, 10, 21, and 121 are prime to each other.

If 53 measures 14363, the remainder, 1431 tens, is (172) a multiple of 53. But the product, 371 tens, is a multiple : therefore, the remainder, 106 hundreds, is also a multiple. Now we have subtracted 53 times 1, 53 times 7 tens and 53 times 2 hundreds, and have no remainder. Hence, the third operation shows that 14363 is a multiple of the primes 53 and 271.

In thus operating, we always multiply the prime, if required, by such a number as will render its unit figure the same as that of the number from which we subtract.

This operation shows that 472903 is the product of the 3 primes 23, 29, and 709.

We must here observe, that we cannot always thus easily obtain satisfactory results, but sometimes find this research exceedingly laborious.

175. The following properties of numbers will be found exceedingly useful in reducing fractions to their lowest terms, in resolving numbers into their prime factors, &c., by which, as well as by other means, they serve greatly to abridge the labour of calculation.

1. Even numbers are divisible by 2.

2. When the sum of the figures of a number is divisible by 3, the number is divisible by 3.

3. Because 4 measures 100, when 4 divides the number expressed by the two right-hand figures of a given number, the given number is divisible by 4.

4. Because 10 is a multiple of 5, and 5 contains itself, when the right-hand figure of a number is 5 or 0, the number is divisible by 5.

5. When an even number is divisible by 3, it is divisible by 6.

6. Because 3.7=21, and 7.1391, when the unit figure is, or of the part on the left, the number is divisible by 7. Again, because 7.15105, and 7.43301, when the number expressed by the two right-hand figures is 5 times the part on the left, or of it, 7 will measure. Also, in the latter case, 43 will measure.

7. Because 8 measures 1000, when 8 measures the number expressed by the three right-hand figures of a number, 8 measures the number.

8. When the sum of the figures of a number is divisible by 9, the number is divisible by 9, (113.) Also, the difference between a number and the same number reversed, is, when not a cypher, divisible by 9. Thus, 8176-67181458, which is a multiple of 9. For, in casting the nines out of each number, we have the same remainder; and, by the subtraction, the remainder is cancelled or destroyed. Hence, in accounts, an error, divisible by 9, may suggest that a number hus, by mistake, been reversed.

9. Because 7.11.13-1001, the numbers 2002, 3003, 4004, &c. to 9009, included: also, 51051, 63063, 324324, 561561, &c., which are easily recognized as multiples of 1001, are divisible by 7, 11, and 13. Also, when the difference between the numbers in the two periods is 7, 11, or 13, or a multiple of any one of them, the number is a multiple of that factor accordingly. Thus 561554 is a multiple of 7: 561539 is a multiple of 11; and 561522 is a multiple of 13.

10. Because 3.17=51, 6.17=102, and 17.59 = 1003, the prime 17 will measure a number; when the unit figure is of the part on the left, or when the number expressed by the two right-hand figures is double the part on the left; also when the number in the first period is 3 times the part on the left, 17 and 59 will measure. Thus, 459 and 2448 are multiples of 17, and 157471 is a multiple of 17 and 59.

11. Because 19.53 1007, either 19 or 53 will measure a number, when the number in the first period is 7 times the part on the left of that period. Thus, 12084 and 113791 are multiples of 19 and 53.

12. Because 1009 is prime, when the number in the first period is 9 times the part on the left, the factors of this part and 1009 are the factors of the number. Thus, in the number 91819, the factors of 91, which are 7 and 13 and 1009 are the only factors. In like manner, when the number in the first period is 13, 19, 21, 31, 33, 39, 49, 51, 61, or 63 times

that in the second, we should recognize the number as a multiple of 1013, 1019, 1021, 1031, 1033, &c., respectively, each of which is a prime number. Thus, in the number 22682, we recognize the factors 2, 11, and 1031.

176. The binomial values of the alts 100, 10000, 1000000, &c.; that is, of those having an even number of ciphers on the right, are 99+1; 9999+1; 999999+1, &c., the nines of each being divisible by 11. Therefore, if we divide each of these alts by 11, the remainder will be 1: consequently, if we divide an alt figure in any of the odd places by 11, the remainder will be that figure. Hence the figures of a number, in the odd places, may be considered as remainders of so many single units, the value of each having been divided by 11.

The alt 10 is (11-1): hence 40 is (11-1)4: or 11+ 11+11+11—4 ; or 11 + 11 + 11 + 7 : that is, in dividing the alt 4 by 11, the remainder is its undecimal complement 7, or what it lacks of 11.

If we divide 4000 by 11, the first division, that of 40 hundreds, gives 7 for remainder; the next, of 70 tens, gives 4; and the last, of 40, gives 7, as at first. Hence, the division of any figure in an even place by 11, gives for remainder the undec. comp. of that figure. The figures of a number in the even places are, therefore, the undec. comps. of the remainders that would arise in dividing the value of each by 11.

177. If, from a number, we subtract the sum of the figures in the odd places, and, to the remainder, add the sum of the figures in the even places, the result will be a multiple of 11. For, we thus take away the remainders that would arise in dividing, by 11, the value of the figures in the odd places: and add the complements of the remainders that we should find in dividing the value of the figures in the even places; and it is plain that the sum of these remainders and their comps. is a multiple of 11.

Wherefore, because addition and subtraction destroy each other, when the sum of the figures in the even places is equal to that of those in the odd places: or, when the difference between these sums is 11, or a multiple of 11, the number is a multiple of 11. The numbers 121, 341, 297, 792, 3454, 18579, 73986, 87969871 are therefore divisible by 11.

As the figures in the odd places may be considered remainders, it is plain that, when their sum is 11, or a multiple of 11, and there are ciphers in the even places, the number is divisible by 11. Thus, the numbers 209, 902, 308, 803, 605,