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Number Theory Level 4

Let { a , b , c , d , e , f , g , h , i } \{a, b, c, d, e, f, g, h, i\} be a permutation of { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 } \{1, 2, 3, 4, 5, 6, 7, 8, 9\} such that gcd ⁡ ( c , d ) = gcd ⁡ ( f , g ) = 1 \gcd(c, d) = \gcd(f, g) = 1 and ( 10 a + b ) c d = e f g . (10a + b)^{\frac{c}{d}} = e^{\frac{f}{g}}. Given that h > i h > i , evaluate 10 h + i 10h + i .


The answer is 65.

Sharky Kesa by Sharky Kesa , 19, United Kingdom

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1 solution

Sharky Kesa
Nov 9, 2017

There are two possible solutions, each of which yield the same values for h h and i i . ( 10 ⋅ 2 + 7 ) 8 3 = 9 4 1 and ( 10 ⋅ 2 + 7 ) 1 4 = 9 3 8 (10 \cdot 2 + 7)^{\frac{8}{3}} = 9^{\frac{4}{1}} \quad \text{and} \quad (10 \cdot 2 + 7)^{\frac{1}{4}} = 9^{\frac{3}{8}} Thus, h = 6 h=6 , i = 5 i=5 , so 10 h + i = 65 10h+i = \boxed{65} .

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