An “Armstrong Number” is an n-Digit Number That is Equal to the Sum of the nth Powers GMAT Problem Solving

Question: 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 ?

1. 2
2. 3
3. 4
4. 5
5. 6

‘An “Armstrong number” is an n-digit number that is equal to the sum of the nth powers of its individual digits’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”.
To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation

Approach Solution 1:
Given:

• An “Armstrong number” is an n-digit number that is equal to the sum of the nth powers of its individual digits
• 153 is an Armstrong number because it has 3 digits and 1^3 + 5^3 + 3^3 = 153

Find Out:

• What is the digit k in the Armstrong number 1,6k4

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
Now, we can take one option from the provided ones:
We see that k must be 3 since 3^4 = 81 and 51 + 10(3) = 81.
Hence, B is the correct answer.

Correct Answer: B

Approach Solution 2:

Given:

• An “Armstrong number” is an n-digit number that is equal to the sum of the nth powers of its individual digits
• 153 is an Armstrong number because it has 3 digits and 1^3 + 5^3 + 3^3 = 153

Find Out:

• What is the digit k in the Armstrong number 1,6k4

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 : Candidates must recognize the question is asking for sum of units digit to be 4
So we will have to 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, let us just look through choices only 3 gives a units digit of 1

1. 2
2. 3
3. 4
4. 5
5. 6

Hence, B is the correct answer.

Correct Answer: B

Approach Solution 3:

It is given that 1,6k4 is a 4-digit number, we can create the equation:
that implies 1^4 + 6^4 + k^4 + 4^4 = 1000 + 600 + 10k + 4
that implies1 + 1296 + k^4 + 256 = 1604 + 10k
that implies k^4 = 51 + 10k
We see that k must be 3 since 3^4 = 81 and 51 + 10(3) = 81.
Correct Answer: B

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