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If x, y, and z are integers, x < y < z and xyz = 30, what is the smallest possible value of y?
A. -6
B. -5
C. -2
D. 2
E. 5
A. -6
B. -5
C. -2
D. 2
E. 5
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02 Feb 2024, 02:46
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If \(x\), \(y\), and \(z\) are integers, \(x < y < z\) and \(xyz = 30\), what is the smallest possible value of y?
A. -6
B. -5
C. -2
D. 2
E. 5
The divisors of 30 are 1, 2, 3, 5, 6, 10, 15, 30, and their negative counterparts.
All \(x\), \(y\), and \(z\) cannot be negative because their product would then be negative. Hence, to obtain the least value of \(y\), we must consider the case when \(x < y < 0 < z\). To minimize \(y\), or equivalently, to maximize the absolute value of \(y\) (\(|y|\)), we need to minimize \(z\). The smallest positive value of \(z\) is thus 1, which leads to \(xy = 30\). Since \(x\) and \(y\) must both be negative, the smallest value of \(y\) is -5 when \(x = -6\).
Answer: B
A. -6
B. -5
C. -2
D. 2
E. 5
The divisors of 30 are 1, 2, 3, 5, 6, 10, 15, 30, and their negative counterparts.
All \(x\), \(y\), and \(z\) cannot be negative because their product would then be negative. Hence, to obtain the least value of \(y\), we must consider the case when \(x < y < 0 < z\). To minimize \(y\), or equivalently, to maximize the absolute value of \(y\) (\(|y|\)), we need to minimize \(z\). The smallest positive value of \(z\) is thus 1, which leads to \(xy = 30\). Since \(x\) and \(y\) must both be negative, the smallest value of \(y\) is -5 when \(x = -6\).
Answer: B
General Discussion
25 Nov 2023, 07:23
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Bunuel
\(30 = 3 * 2 * 5\)
We can use the answer options to solve this question.
Given: x < y < z
Inference: On a number line, \(x\) lies to the left of \(y\) and \(z\) lies to the right of \(y\). The relative positions of x, y, and z are shown below -
----------- \(x\) ----------- \(y\) ----------- \(z\) -----------
Answer choice elimination
We can start with the lowest value in the options
A. \(-6\)
If y = -6, x*z = -5
Possible values of x and z
- x = -1 ; z = 5
- x = 5 ; z = -1
However, given the constraint on the placement of x, y, and z on the number line, none of the above combinations are possible. Eliminate A.
B. \(-5\)
If y = -5, x*z = -6
Possible values of x and z
- x = -1 ; z = 6
- x = -6 ; z = 1
One of the two possibilities matches the constraints given -
----------- \(-6\) ----------- \(-5\) ----------- \(1\) -----------
Hence, the smallest possible value of \(y = -5\)
Option B
