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How many multiples of 9^3 are greater than 9^4 and less than 9^5?

 Feb 13, 2023
 #1
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a=listfor(n, 7290/729, 58320/729, n*729); printa, count a

 

(7290, 8019, 8748, 9477, 10206, 10935, 11664, 12393, 13122, 13851, 14580, 15309, 16038, 16767, 17496, 18225, 18954, 19683, 20412, 21141, 21870, 22599, 23328, 24057, 24786, 25515, 26244, 26973, 27702, 28431, 29160, 29889, 30618, 31347, 32076, 32805, 33534, 34263, 34992, 35721, 36450, 37179, 37908, 38637, 39366, 40095, 40824, 41553, 42282, 43011, 43740, 44469, 45198, 45927, 46656, 47385, 48114, 48843, 49572, 50301, 51030, 51759, 52488, 53217, 53946, 54675, 55404, 56133, 56862, 57591, 58320)= 71 multiples of 9^3 are greater than 9^4 and less than 9^5.

 Feb 13, 2023
 #2
avatar+2667 
+1

Note that \(9^4 = 9 \times 9^3\) and \(9^5 = 81 \times 9^3\)

 

We have the inequality \(9 \times 9^3 < x \times 9^3 <81 \times 9^3\), which is equal to \(9

 

The possible cases for x are 10, 11, 12, ... 80. 

 

Thus, there are a total of \(80 - 10 + 1 =\color{brown}\boxed{71}\) cases.

 Feb 14, 2023

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