It quickly becomes apparent that 174 is much too large, so n must be 144.The value of the n is 144.
Taking the given equation modulo 2,3, and 5, respectively, we have
n⁵≡(mod 2)
n⁵≡(mod 3)
n⁵≡( mod 5)
By either Fermat's Little Theorem (FLT) or inspection, we get
n≡(mod 2)
n≡(mod 3)
n≡( mod 5)
By either the Chinese Remainder Theorem (CRT) or inspection, we get n≡ 24 ( mod 30)
It is clear that n>133, so the possible values for n are 144,174,204,...Note that
n⁵ = 133⁵ + 110⁵ + 84⁵ + 27⁵
n⁵ < 133⁵ + 110⁵ +( 84 + 27)⁵
n⁵ =133⁵ + 110⁵ + 111⁵
n⁵ < 3.133⁵
From which (n/133)⁵ < 3
If n>= 174 then,
(n/133)⁵ > 1.3⁵
(n/133)⁵ = 3
which arrives at a contradiction. Therefore, we conclude that n=144
Learn more about Fermat's Little Theorem here:
brainly.com/question/8978786
#SPJ4
