Mathematical_Recreations-Kraitchik-2e

62 Mathematical Recreations
person to divide his number by 42, 48, 56, and 63, and to tell
you the respective remainders, say a, b, c, d. Then
N == 1,008 - (96a + 21b + 90c + 800d) (m = 1,008).
To show this, let
N = 42p+ a = 48q+ b = 56r+ c = 63s+d. Then
96N = 1,008·4p + 96a
21N = 1,008·q + 21b
90N = 1,008·5r + 90c
800N = 1,008·50s+ 800d
1,007N = 1,008·S + (96a+ 21b+ 9Oc+ 800d).
1,2,3,4,9,10,11,12,17,18,19,20,25,26,27,28
5,6,7,8,13,14,15,16,21,22,23,24,29,30,31,32
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3,7,11,15,19,23,27,31,33,37,41,45,49,63,57,61
2,6,10,14,18,22,26,30,36,40,44,48,52,56,60,64
• • II II
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13,14,15,16,25,26,27,28,45,46,47,48,57,58,59,60
9,10,11,12,29,30,31,32,41,42,43,44,61,62,63,64
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1 234 5 6 7 8
9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24
25 26 27 28 29 20 31 32
33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56
57 58 59 60 61 62 63 64
FIGURE 3. Window Reader's Cards for Numbers 1 to 64.