BTGModeratorVI wrote: ↑

Wed Oct 07, 2020 7:23 am

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

Answer: B
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

BTGModeratorVI wrote: ↑

Wed Oct 07, 2020 7:23 am

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

Answer: B
Source: official guide

Solution:

Since 1,6k4 is a 4-digit number, we can create the equation:

1^4 + 6^4 + k^4 + 4^4 = 1000 + 600 + 10k + 4

1 + 1296 + k^4 + 256 = 1604 + 10k

k^4 = 51 + 10k

We see that k must be 3 since 3^4 = 81 and 51 + 10(3) = 81.

Answer: B