Algebra for Schools And Colleges

Cover Algebra for Schools And Colleges
Algebra for Schools And Colleges
Newcomb Simon
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7 goes into QQ 9 times and 3 over. Hence this means 66 = 7. 9 + 3, or 66-7. 9=1:3.
1 1 DIVISIBILITY OF NUMBERS. 241 333. Peoblem. To find the greatest common divisor of two nuwyhers.
Let m and n be the numbers, and let m be the greater.
1. Divide m by n. If the remainder is zero, n will be the divisor required, because every number divides itself. If there IS a remainder, let q be the quotient and r the remainder.
Then m — nq^ r.
Let d be the common divisor required. Because m and n, are both d
...ivisible by ^, m — nq must also be divisible by d (Theorem II). Therefore, ^r is divisible by d. Hence every common divisor of m and n is also a common divisor of n and r. Conversely, because m :=z nq + r, every common divisor of n and r is also a divisor of m. There- fore, the greatest common divisor of m and n is the same as the greatest common divisor o^ n and r, and we proceed with these last two numbers as we did with in and n, 2. Let r go into n q' times with the remainder t\ Then n = rq' -{- r\ 'Or n ^rq' — /.

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