CASTING OUT ELEVENS. 88. When a number expressed by a digit in an odd place is divided by 11, the remainder is equal to that digit; and a number expressed by a digit in an even place, lacks that digit of being a multiple of 11. Therefore, if a number expressed by two figures is divided by 11, the remainder equals the digit in the odd place minus the digit in the even place. Thus, in 4500, 500 represented by 5 in the third place, divided by 11, has a remainder of 5. 4000 represented by 4 in the fourth place, lacks 4 of being a multiple of 11. 541. 4500 divided by 11 has a remainder of 1. If the digit in the even place is greater than that in the odd place, it cannot be subtracted, so we add one 11 to it, and then proceed to subtract. Hence, any number divided by 11 will have a remainder equal to the sum of the digits in the odd places minus the sum of those in the even places. From these principles, we deduce proofs of the fundamental processes by casting out 11's, similar to the proofs by casting out 9's. NOTE.- —The following examples are the same as those used to illustrate the proofs by casting out 9's. EXAMPLES. 1. Prove that 3523, 6414, 1894, and 2129 = 13960. 3523, 8. 5 = 3 6414, 8. 7 = 1 10= 2 4 = 6 13960, 10-9=1. Exc. = 1. 12, Exc. = 1. SOLUTION. The sums of the digits in the odd places minus those in the even places in the various addends are 3+1+2+6 12; excess of 11's in 12 = 1. The sum of the digits in the odd places of 13960 minus those in the even places = 1; ex cess of 11's = 1. Hence, the answer is probably correct. 2. Prove that 15964 - 9432 6532. -11=3(+11) 9, Exc. 15964, 14 9432, 6532, 6(+11)-12=5 62, 2(+11)-6-7 2604 10-2=8, Exc. 7(+11)-99, Exc. hend the sum of the digits in the odd places is 6, and in the even places 12. Since 12 cannot be subtracted from 6, we add one 11 to 6, making 17, and 17 — 12 = 5. We subtract the results; but since 5 cannot be subtracted from 3, we add one 11 to 3, making 14, and 14-5-9. In the same way we find the excess of 11's in 6532 to be 9. Since the two excesses of 11 agree, the answer is probably correct. 3. Prove that 62 × 42 63 Exc. 8 = SOLUTION. In the minuend the sum of the digits in the odd places is 14, and in the even places 11. Their difference is 3. In the subtra = 4. Prove that 207 23 = 9. SOLUTION. 23)207 (9 in quotient, 9. 2646. SOLUTION. The excess of 11's in 62 is 7; in 42, 9. Their product 63 has an excess of 8. In 504, the excess of 11's is 8. Since the two excesses are the same, the answer is probably correct. - Excess of 11's in divisor is 1, Their product is 9. The excess Exc. 3-2-1 9 9 of 11's in the dividend is 9. Since the excesses are the same, the answer is probably correct. RULE. -I. In Addition, subtract the sum of the digits in the even places from the sum of those in the odd places in each addend. Add the results and find the excess of 11's in their sum. If this agrees with the excess of 11's in the answer, the answer is probably correct. II. In Subtraction, subtract the excess of 11's in subtrahend from that in minuend. Compare with answer. III. In Multiplication, cast out 11's from the product of the excess of 11's in the multiplicand and multiplier, and compare with answer. IV. In Division, find the product of excess of 11's in the divisor and quotient; cast out the 11's. Compare with excess in dividend. FACTORS. DIVISIBILITY OF NUMBERS. 89. An Even Number is a number that is exactly divisible by 2. All numbers whose unit figure is 0, 2, 4, 6, or 8, are even. 90. An Odd Number is a number that is not exactly divisible by 2. All numbers whose unit figure is 1, 3, 5, 7, or 9, are odd. 91. A Prime Number is a number that has no integral factors except unity and itself. Thus, 2, 3, 5, 11, 23, etc., are prime numbers. 2 is the only even prime number. 92. A Composite Number is a number that has other integral factors besides unity and itself. Thus, 21 is a composite number, since 21 = 7 x 3. 93. The Factors of a number are the numbers whose product is the given number. Thus, 7 and 8 are factors of 56; 3, 4, and 7, of 84. A Prime Factor is a prime number used as a factor. The prime factors of a number are also the prime divisors of it. 94. Numbers are prime to each other when they have no common integral factors, or divisors. Thus, 9 and 14, 16 and 25, are prime to each other. 95. An Exact Divisor of a number is one that will divide that number without a remainder. Thus, 6 is an exact divisor of 48, and 9 is an exact divisor of 72. The exact divisors of a number are also the factors of that number. An exact divisor of a number is sometimes called the measure of that number. When a number is a factor, or divisor, of each of two or more numbers, it is called a common factor, or divisor of those numbers. 2 is an exact divisor of all even numbers. 3 is an exact divisor of any number, the sum of whose digits is divisible by 3. Thus, 3 is an exact divisor of 696, 3735, 840. 4 is an exact divisor if its two right hand figures are ciphers, or express a number divisible by 4. Thus, 4 is an exact divisor of 200, 756, 1284. 5 is an exact divisor of every number whose unit figure is 0 or 5. Thus, 5 is an exact divisor of 20, 955, and 2840. 6 is an exact divisor of an even number if the sum of its digits is divisible by 3. Thus, 6 is an exact divisor of 549, 678, 399. 8 is an exact divisor when it will exactly divide the hundreds, tens, and units of a number. Thus, is an exact divisor of 1728, 5280, and 213560. 9 is an exact divisor when it will exactly divide the sum of the digits of a number. Thus, in 2486790, the sum of the digits 2 + 4 +8+ 6+ 7+9+0 = 36, and 36 ÷ 9 = 4. 10 is an exact divisor when 0 occupies units' place. 11 is an exact divisor when the difference between the sum of its even digits and the sum of its odd digits is exactly divisible by 11, or when it is 0. Thus, 11 is an exact divisor of 4554, 91322. 100 is an exact divisor when 00 occupy the places of units and tens. 1000 is an exact divisor when 000 occupy the places of units, tens, and hundreds, etc. A composite number is an exact divisor of any number, when all its factors are exact divisors of the same number. Thus, 2, 2, and 3 are exact divisors of 12; so also are 4 (= 2 × 2) and 6 ( = 2×3). An even number is never an exact divisor of an odd number. If an odd number is an exact divisor of an even number, the quotient will be an even number. If an odd number is an exact divisor of an even number, twice that odd number is also an exact divisor of the even number. Thus, 7 is an exact divisor of 42; so also is 7 × 2, or 14. 1235 For reference, and to aid in determining the prime factors of composite numbers, we give the following table: 3 TABLE OF PRIME NUMBERS FROM 1 TO 1000. 59 139 233 337 439 557 61 149 239 347 443 563 659 67 151 241 349 449 569 661 71 157 251 353 457 571 673 73 163 257 359 461 577 677 809 919 11 79 167 263 367 463 587 683 811 929 13 83 173 269 373 467 593 691 821 937 17 89 179 271 379 479 599 701 823 941 19 97 181 277 383 487 601 709 827 947 23 101 191 281 389 491 607 719 829 953 283 397 613 727 839 967 29 103 193 31 107 197 293 401 617 733 853 971 307 409 619 739 419 37 109 199 631 743 421 523 641 751 431 541 757 53 137 229 331 643 433 231 77 11 1 499 503 509 521 FACTORING. EXAMPLES. 653 769 883 773 887 787 907 797 911 96. To resolve any composite number into its prime factors. 857 977 859 983 863 991 877 997 1. What are the prime factors of 2772 ? OPERATION. SOLUTION. We divide the given number by 2, the 22772 least prime factor, and the result by 2; this gives an 2 1386 odd number for a quotient, divisible by the prime factor 3, and the quotient resulting from this division 3 693 is also divisible by 3. The next quotient, 77, we 3 7 11 divide by its least prime factor, 7, and obtain the quotient 11; this being a prime number, the division cannot be carried further. The divisors and last quotient, 2, 2, 3, 3, 7, and 11, are all the prime factors of the given number, 2772. |