By A. Adrian Albert
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7• Solution The integer P ... 9 has nine factors, the product of the first seven of which is P ... 7• Then P •• 9 = P ... 7(n - 7)(11. - 8). Here the final factor is 11. - 'I' + 1 where 'I' = 9. Then P ... 7(n - 7)(11. - 8) = 42P... 7 , 7)(11. - 8) = 42. The pro 6,720; P 9• S = 15,120. Hence 11. = 9, 'I' = 5. 'I' ORAL EXERCISES Express the ·first of the following numbers in terms of the second (using products or quotients): · 24 NATURAL NUMBERS L 81,61 2. 61,81 3. 101, 91 4. 101,81 6. 91, 61 6. 61, 71 7. P •. G, P •. 4 8. •- l 9. "-l ,10. s, 0.... & EXERCISES 1. Simplify the following by writing the result as an integral multiple of nl for n as large as possible: (a) (b) (0) (d) 81 - 4' 3' 91 8' 51 12' 81 - (6) 3' 71 - 2 . 61 (f) 4· 71 - 6 . 51 (g)'3161 71 (h) (41)2 2 . I and II as properties of integers. We state this property as a law. XI. LAW OF DIVISION. If a andp are any two numbers such that P is not zero, there is a unique number 'Y in the number system such that a == fJ'Y. We shall call 'Y the quotient of a by P and write a -• . == P 54 REAL AND COMPLEX NUMBERS [CRAP. m Then every rational number is the quotient of two rational numbers which happen also to be integers. Thus the rational number system may be thought of as resulting from the system of integers by adjoining to it all quotients of integers.