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Therefore, 2x2x2x7x13=23 x 7 x 13 are the prime fac. tors of 728. 2. Resolve 812 into its prime factors.

Ans. 22 X 7 X 29. 3. What are the prime factors of 978.

Ans. 2x3 x 163. 4. What are the prime factors of 1011?

Ans. 3 x 337, 5. What are the prime factors of 100?

Ans. 22 x 52 6. What are the prime factors of 8975?

7. What are the prime factors of 808?

8. What are the prime factors of 707?

9. What are the prime factors of 1118 ?

10. What are the prime factors of 1098 ?"

7. As we shall make so frequent use of prime numbers, we will give a table of some of the lowest primes.

TABLE OF PRIME NUMBERS. 11 131/311)50317191 941) 1163) 1423 1619|1877) 2129 2 137 313 509 727 947 1171 1427 1621 1879 2131 3 139 317 521 733 953) 1181 1429 1627 1889 2137 5 149 331 523 739 967 1187 1433 1637 1901 2141 7 151 337 541 743 971 1193 1439 1657 1907 2143 11 157 347 547 751) 977 1201 1447 1663 1913 2153 13 163 349 557 757) 983 1213 1451 1667 1931 2161 171 1671 353 563) 7611 991 1217/ 1453 16691933 2179 19 173 359 5691 7691 997) 1223 1459) 1693 1949) 2203 23 179367 571 773 1009 1229 1471 1697 1951 2207 29 181/ 373577) 787) 1013) 1231 1481 16991973) 2213 31 191 379 587 797) 1019 1237 1483 1709 1979 2221 37 1931 383 593 809) 1021) 12491487) 172119871 2237 411197 389/599 811 1031 1259 1489 1723) 1993 2289 43 199) 397 601 821 1033 1277 1493 1733 1997 2243 47 211 401 607 823) 1039 1279 1499 1741 1999 225). 53 223 409 613 827 1049 1283 1511 1747/ 2003 2267 59 227 419 617 829 1051 1289 1523 1753 2011 2269 611 229 421) 61918391061 1291 1531 1759 2017 2273 67 233 431|631 853 1063 1297 1543 1777 2027 2281 71 239) 433 641 857 1069 1301 1549 1783 2029 2287 731 2411 4391 6431859) 1087 1303 1553 178712039 2293 79 251 443 647 863 1091 1307 1559 1789 2053 2297 83 2571 449653 877| 1093) 1319) 1567) 18011 2063 2309 89 263 457 659 881 1097 1321 1571 1811 2069 2311 97 269| 461 661 883) 11031 1327) 1579 1823 2081 2333 101 271 463 673 887 1109 1361 1583 1831 2083 2339 103\277) 467 677 9071117 1367) 1597| 18471 2087! 2341 107 281 479 583 911 1123 1373 1601 1861 2089 2347 109/283148716911 919) 1129 1381 1607) 18672099 2351 113 293 491 701929 1151 1399 1609 1871 2111 2357 127 307 499 709 937) 1153 1409 1613 1873 2113 2371 . 8. Suppose we wish to know whether the numbers 204 and 468 have a common factor ; we proceed as follows: We decompose them into their prime factors, and thus obtain 204=22 X 3 X 17, and 468=22 X 32 X 13. Here we see that 22 X3 is common to both the numbers 204 and 468.

Hence, to find the greatest factor which is common to two or more numbers, or, as generally expressed, to find the great. est common measure of two or more numbers, we have this

RULE. Resolve the numbers into their prime factors, (by Rule under Art. 6.) Then select such of the primes as are common to all the numbers, multiply them together, and the product will give the greatest common measure.

Examples. 1. What is the greatest common measure of 1326, 3094 and 4420 ?

These numbers, when resolved into the prime factors, become

1326=2X3 X 13 x 17
309452x 17 x 13 x 17

4420=22 X 5 X 13 x 17 The factors which are common are 2, 13, and 17; there. fore the greatest common measure is 2X13 x 17=442.

2. What is the greatest common measure of 556,672, and 840 ?

Ans. 22=4. 3. What is the greatest common measure of 110, 140, and 680?

Ans. 2x5=10. 4. What is the greatest common measure of 255, and 532 ?

Ans. They have none. 5. What is the greatest common measure of 375, 408, and 922 ?

Ans. They have none.

9. We may also find the greatest common measure of two numbers by the following . ' '

, RULE: Divide the greater by the less, then divide the divisor by the remainder, and thus continue to divide the preceding divisor by the last remainder, until there is no remainder. The last divisor will be the greatest common measure.

Examples. ''; 1. What is the greatest common measure of 360, and 630 ?

OPERATION. I
360)630(1

360
270)360(1

270
90)270(3
270

0 Hence the greatest common measure is 90.

2. What is the greatest common measure of 922, and 408?...

Ans. 2. 3. What is the greatest common measure of 1825, and 2555 ?

Ans. 365. 4. What is the greatest common measure of 124, and 682?

Ans. 62. 5. What is the greatest common measure of 296, and 407 ?

Ans. 37. - 6. What is the greatest common measure of 404, and 364?.

Ans. 4. 7. What is the greatest commmon measure of 506, and 308 ?

8. What is the greatest common measure of 212, and 416 ? '

9. What is the greatest common measure of 74, and 84 ? 10. Suppose we wish to know what is the least number which will divide by 215 and 460 ; we proceed as follows : We decompose them into their prime factors, and thus ob. tain 215=5 X 43, 460=22 X5 X 23. Hence, we see that 22 X 5 X 23 X 43=19780, is the least number' which can be divided by 215 and 460.

Hence, to find the least number which will divide by two or more numbers, or as generally expressed, to find the least common multiple, we have this

RULE. Resolve the numbers into their prime factors (by Rule under Art. 6,) Select all the different factors which occur, observing that, when the same factor has different powers, to take the high. est power. The continued product of the factors thus selected will give the least common multiple.

Examples. 1. What is the least common multiple of 12, 16, and 24 ? These numbers resolved into their prime factors give

-]2=22 X3

16=24.

-24=23 X 3 Therefore 24 x 3=48 is the least multiple required.

2. What is the least common multiple of 9, 12, 16, 20, and 35 ?

Ans. 5040. 5. What is the least common multiple of 7, 13, 39, and 84?

Ans. 1092. 4. What is the least common multiple of the nine digits ?

Ans. 2520. 5. What is the least common multiple of 3, 5, 7, 12, 15, 18, and 35?

Ans. 1260. 6. What is the least common multiple of 100, 109, 463, and 900 ?

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