An “Armstrong number” is an n-digit number that is equal to the sum

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BTGModeratorVI: Legendary Member; Posts: 1223; Joined: Sat Feb 15, 2020 2:23 pm; Followed by:1 members

An “Armstrong number” is an n-digit number that is equal to the sum

by BTGModeratorVI » Wed Oct 07, 2020 7:23 am

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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

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Source: Beat The GMAT — Problem Solving | Wed Oct 07, 2020 7:23 am

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pm; Location: Vancouver, BC; Thanked: 5254 times; Followed by:1268 members; GMAT Score:770

Re: An “Armstrong number” is an n-digit number that is equal to the sum

by Brent@GMATPrepNow » Thu Oct 08, 2020 5:43 am

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

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

Re: An “Armstrong number” is an n-digit number that is equal to the sum

by Scott@TargetTestPrep » Mon Oct 12, 2020 4:18 am

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

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