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

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guddo
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The sum and product of k positive integers are 12 and 81, respectively [#permalink] New post   28 Jan 2024, 08:38
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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

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D
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Bunuel
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Re: The sum and product of k positive integers are 12 and 81, respectively [#permalink] New post   28 Jan 2024, 08:41
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guddo wrote:
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

Show Spoiler
Attachment:
2024-01-28_19-26-18.png


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.
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Re: The sum and product of k positive integers are 12 and 81, respectively [#permalink] New post   28 Jan 2024, 10:05
guddo wrote:
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

Show Spoiler
Attachment:
2024-01-28_19-26-18.png


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
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The sum and product of k positive integers are 12 and 81, respectively [#permalink] New post   04 Feb 2024, 04:28
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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 )
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The sum and product of k positive integers are 12 and 81, respectively [#permalink] New post   12 Feb 2024, 06:56
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.
 ­­
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Re: The sum and product of k positive integers are 12 and 81, respectively [#permalink] New post   12 Feb 2024, 07:20
@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
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Re: The sum and product of k positive integers are 12 and 81, respectively [#permalink] New post   12 Feb 2024, 07:27
 
guddo wrote:
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

Show Spoiler
Attachment:
2024-01-28_19-26-18.png

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

Thus, \((\frac{12}{k})^k = 3^4\)­

Hence, \(k = 4\), Answer must be (D)
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Re: The sum and product of k positive integers are 12 and 81, respectively [#permalink]
12 Feb 2024, 07:27
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