More problems in 2016 part 2

Number Theory Level 4

For natural numbers : a ,b ,c and d (not necessarily distinct) ; Following conditions are true.

a + b + c + d = 18 3 a b c d = 525 } \left.\begin{matrix} &a+b+c+d=18 & \\ & & \\ & 3abcd=525 & \end{matrix}\right\}

Then find the value of :

a 3 + b 3 + c 3 + d 3 a^3 + b^3 + c^3 +d^3


The answer is 594.

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1 solution

Shaun Leong
Dec 29, 2015

3 a b c d = 525 3abcd=525 a b c d = 175 = 5 2 ∗ 7 abcd=175=5^2*7

a,b,c and d are natural numbers, their product is 175 and their sum is 18 < 25 18 <25 . Hence they are single-digit factors of 175 and ( a , b , c , d ) = ( 1 , 5 , 5 , 7 ) (a,b,c,d)=(1,5,5,7)

This satisfies all 3 conditions. The answer is therefore 1 3 + 5 3 + 5 3 + 7 3 = 594 1^3+5^3+5^3+7^3=\boxed {594}

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Perfect! Did it the same way!+1!

Rishabh Tiwari - 4 years, 12 months ago

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