An “Armstrong number” is an \(n\)-digit number that is equal to the sum of the \(nth\) powers of its individual digits.

[Gmat_mission]

An “Armstrong number” is an \(n\)-digit number that is equal to the sum of the \(nth\) powers of its individual digits. For example, 153 is an Armstrong number because it has 3 digits and \(1^3 + 5^3 + 3^3 = 153.\) What is the digit \(k\) in the Armstrong number \(1,6k4 ?\)

A. 2
B. 3
C. 4
D. 5
E. 6

[spoiler]OA=B[/spoiler]

Source: Official Guide

[Brent@GMATPrepNow]

An “Armstrong number” is an \(n\)-digit number that is equal to the sum of the \(nth\) powers of its individual digits. For example, 153 is an Armstrong number because it has 3 digits and \(1^3 + 5^3 + 3^3 = 153.\) What is the digit \(k\) in the Armstrong number \(1,6k4 ?\)

A. 2
B. 3
C. 4
D. 5
E. 6

[spoiler]OA=B[/spoiler]

Source: Official Guide

1,6k4 is a 4-digit number.
So, 1⁴ + 6⁴ + k⁴ + 4⁴ = 16k4
Evaluate: 1 + 1296 + k⁴ + 256 = 16k4
Simplify: 1553 + k⁴ = 16k4

Whatever k is, it must be the case that the UNITS digit of k⁴ is 1, so that 1553 + k⁴ = 16k4
Test some values..
k = 1: 1⁴ = 1, so we get: 1553 + 1 = 1554 NO GOOD
k = 3: 3⁴ = 81, so we get: 1553 + 81 = 1634 WORKS!!

Answer: k = 3

Cheers,
Brent