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| A Seven Digit Problem (Posted on 2006-09-07) |
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Determine a 7-digit number ABCDEFG where each letter denotes a digit (not all digits must be distinct) and A≠0, such that:
ABCDEFG/FGABCDE=5/12 and EFGABCD/DEFGABC=51/29
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Submitted by K Sengupta
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Solution:
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The required number is 1673640.
EXPLANATION:
By the conditions of the problem:
EFGABCD/ DEFGABC = 51/29;
Or, (10* EFGABC + D)/ ( 1,000,000*D + EFGABC) = 51/29;
Or, EFGABC/ D = 213389 (upon simplification)
Or, EFGABC = 213389*D ---------------(#)
Also, ABCDEFG/ FGABCDE = 5/12
Or, (100*ABCDE + FG)/ (100,000*FG + ABCDE) = 5/12
Or, ABCDE/FG = 2092/5
Or, FG = 5*y (say), giving, ABCDE = 2092*y---------(##)
By trial and error, we observe that a solution to the relationships (#) and (##) is possible only when, D = 3 and y =8, giving EFGABC= 640167; D=3; ABCDE = 16736; and FG = 40; so that, the required number is 1673640.
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Subject |
Author |
Date |
 | Solution | Penny | 2006-09-07 10:09:24 |
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