The sum and product of k positive integers are 12 and 81, respectively

Question

The sum and product of k positive integers are 12 and 81, respectively. If the k positive integers are all equal to each other, what is the value of k? A. 9 B. 6 C. 5 D. 4 E. 3 2024-01-28_19-26-18.png

Bunuel · Answer

Assuming each integer equals x, we have:

x + x + ... (k times) = 12, which gives xk = 12
x * x * ... (k times) = 81, which gives x^k = 81

Since both x and k are positive integers, x^k = 81 implies that (x, k) can be either (3, 4) or (9, 2). However, only (3, 4) satisfies xk = 12. Therefore, k = 4.

Answer: D.

gmatophobia · Answer

Assume that the value of each number is x

k*x = 12

x^k = 81

81 = 9 * 9 =  9^2  → k = 2; x = 9 → k*x = 18 ⇒  Eliminate 
or
81 = 3*3*3*3 =  3^4  → k = 4; x = 3 → k*x = 12 ⇒  Correct

Therefore k = 4

Option D

sachi-in · Answer

Sum of k equal numbers ( let's assume the Integer is i ) is : k * i
Product : i ^ k

now
Product:    12 = 3 * 4 or 2 * 6 or 1 * 12 etc.
Sum:         81 = 9 ^ 2 or 3 ^ 4

Only the pair (3 , 4) satisfies both the product and the sum.
But the important thing to note is that the Integer is 3 not 4 and K = 4. ( this subtle mistake will have to be avoided and is the reason for most of the wrong answers )

GmatKnightTutor · Answer

The sum and product of k positive integers are 12 and 81, respectively. If the k positive integers are all equal to each other, what is the value of k?

A. 9
B. 6
C. 5
D. 4
E. 3

(Integer value) (Number of integers) = 12 ----> This is equivalent to the SUM of the integers because the integers are all the same value.
(Integer value)^(Number of integers) = 81 ----> This is equivalent to the PRODUCT of all the integers because the integers are all the same value.

In a sense, you could think of it as

xy = 12
x^y = 81

Noting that all the variables have to be positive integers, that they have to be factors of 12, and how "3" or "9" (i.e. 3^4 or 9^2) can turn into 81 could be helpful.

(3) (4) = 12
(3)^4 = 81

The NUMBER of integers is 4.

Kinshook · Answer

@guddoGiven: The sum and product of k positive integers are 12 and 81, respectively.
Asked: If the k positive integers are all equal to each other, what is the value of k?
Let all of k positive integers be x.
kx = 12
x^k = 81=3^4=9^2

If k=2; x=9; kx=18 and not 12
If k=4; x=3; kx = 12

IMO D

Abhishek009 · Answer

kx = 12  So,  x = \frac{12}{k} 
 k^x = 81

Thus,  (\frac{12}{k})^k = 3^4

Hence,  k = 4 , Answer must be (D)

gmatprepguy1 · Answer

since k positive integers  are all equal to each other , why wouldn't it be written as k^x and k = 3? I see how this invalidates the sum aspect but struggling to see why its written as x^k

Bunuel · Answer

The question says there are k numbers, and they’re all equal to each other.

Suppose k = 4 and each number is 3:

The sum is 3 + 3 + 3 + 3 = 4 * 3 = 12

The product is 3 * 3 * 3 * 3 = 3^4 = 81

So we write the product as x^k because we're multiplying the same number (x = 3) a total of k = 4 times.

If we wrote k^x = 4^3 = 64, that would mean 4 appears 3 times — not what the question says. The roles of x and k would be reversed.

ablatt4 · Answer

Translate literally and pay special attention to operations 
kx=12
x^k=81

12=2^2*3
81=3^4
3^4=81
3*4=12

4 D

The sum and product of k positive integers are 12 and 81, respectively

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The sum and product of k positive integers are 12 and 81, respectively

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