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

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Senior Manager
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An “Armstrong number” is an n-digit number that is equal to the sum of  [#permalink]

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21 Sep 2019, 13:21
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74% (02:53) correct 26% (02:55) wrong based on 186 sessions

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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

PS36302.01
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GMAT 1: 760 Q50 V42
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Re: An “Armstrong number” is an n-digit number that is equal to the sum of  [#permalink]

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22 Sep 2019, 04:15
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gmatt1476 wrote:
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

PS36302.01

$$1^4+6^4+k^4+4^4=16k4$$

The units digits of $$1^4$$, $$6^4$$, and $$4^4$$ are 1, 6, and 6, respectively. Since 1+6+6=13, the units digit of $$k^4$$ must be 4-3=1.

The only answer choice whose 4th power has a unit digit of 1 is B.

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Re: An “Armstrong number” is an n-digit number that is equal to the sum of  [#permalink]

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21 Sep 2019, 21:00
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1⁴+6⁴+k⁴+4⁴=1604+10k
1+1296+k⁴+256=1604+10k
k⁴-10k=1604-1553=51
By replacing the options with k, 3 would be the answer.
3⁴-30=51
Option B

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Re: An “Armstrong number” is an n-digit number that is equal to the sum of  [#permalink]

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22 Sep 2019, 01:02
digit 16k4 ; 1^4+6^4+k^4+4^4
so test with each option value
at k=3 we get 1634 =1^4+6^4+3^4+4^4
IMO B

gmatt1476 wrote:
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

PS36302.01
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Joined: 03 Apr 2015
Posts: 252
GMAT 1: 650 Q49 V31
GPA: 3.59
An “Armstrong number” is an n-digit number that is equal to the sum of  [#permalink]

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01 Dec 2019, 02:07
1
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 ?

this question can be solved just under 30 seconds or may eat up 2 mins as the timer suggests

According to the statement : any AMis given by an n-digit number that is equal to the sum of the nth powers of its individual digits
so for 16k4

1^4 +6^4 +K^4+ 4^4=16k4 : recognize the question is asking for sum of units digit to be 4
so find units digit of each expression and then add up to 4
units digit of 1^4 =1
units digit of 6^4 = 6
units digit of 4^ 4 = 6
so if we add unit digit of each expression we get 13
therefore 3 at units digit + units digit of K^4 = 4

therefore units digit of k^4 has to be 1
now just look through choices only 3 gives a units digit of 1
A. 2
B. 3
C. 4
D. 5
E. 6

though the explanation is big but if you just understand what i did its just a matter of 40 seconds max to get through this daunting question and save time

please give KUDOS if you liked my solution
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Re: An “Armstrong number” is an n-digit number that is equal to the sum of  [#permalink]

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22 Feb 2020, 21:39
1
Top Contributor
gmatt1476 wrote:
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

PS36302.01

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!!

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Re: An “Armstrong number” is an n-digit number that is equal to the sum of   [#permalink] 22 Feb 2020, 21:39
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