SOLUTION: In every numeration system (base x) there is a three-digit integer that is (x+1) times the sum of its digits. In base ten the number is 198. In base three the number is 121. Find t
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Question 1133635: In every numeration system (base x) there is a three-digit integer that is (x+1) times the sum of its digits. In base ten the number is 198. In base three the number is 121. Find the number in base seven that has this property. Answer by greenestamps(13198) (Show Source):
Let the 3-digit number in base 7 be abc (the 3-digit number -- not the product of a, b, and c).
In base 10, the value of the 3-digit base 7 number abc is 49a+7b+c; 8 times the sum of the digits is 8(a+b+c). So we are looking for a, b, and c such that
The largest digit in base 7 is 6; and a can't be 0 because it is the leading digit. Logical analysis then tells us a has to be 1; then for 41a-7c to be a digit less than 7, c has to be 5; that makes b = 41-35 = 6.
ANSWER: The 3-digit number in base 7 for which the number is 8 times the sum of its digits is 165.
