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01 Aug 2023, 01:30
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
62% (01:15) correct
38%
(01:34)
wrong
based on 186
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If \(\frac{st}{u}\) is an integer, which of the following must also be an integer?
A. \(stu\)
B. \(\frac{3s^2t^2}{u^2}\)
C. \(\frac{su}{t}\)
D. \(\frac{tu}{s}\)
E. \(\frac{s}{tu}\)
A. \(stu\)
B. \(\frac{3s^2t^2}{u^2}\)
C. \(\frac{su}{t}\)
D. \(\frac{tu}{s}\)
E. \(\frac{s}{tu}\)
15 Aug 2023, 09:21
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Given \(\frac{ {st}}{u }\) is an integer.
From the options we can find a similar expression
\(\frac{{3*s^2*t^2} }{{u^2}} = 3*{(\frac{{st}}{{u}})}^2\)
We know if \(n\) is an integer then \(n^2\) is also an integer and and integer multiplied by another integer is also an integer.
So B
From the options we can find a similar expression
\(\frac{{3*s^2*t^2} }{{u^2}} = 3*{(\frac{{st}}{{u}})}^2\)
We know if \(n\) is an integer then \(n^2\) is also an integer and and integer multiplied by another integer is also an integer.
So B
14 Feb 2026, 09:52
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This is a great question that tests your understanding of Number Properties, specifically how integer relationships behave when you manipulate expressions.
The key concept here is recognizing what "must be true" versus what "could be true." We're told that st/u is an integer — let's call it n. So st/u = n, which means st = nu.
Step 1: Understand what we know
Given: st/u = integer
Step 2: Evaluate each answer choice
Let's test with concrete numbers. Say s=2, t=3, u=6. Then st/u = 6/6 = 1 ✓ (integer)
(A) stu = 2×3×6 = 36. But wait — try s=1, t=2, u=2. Then st/u = 2/2 = 1 ✓, but we can't say stu must be an integer based on our given condition alone.
(B) 3s2t2/u2 — Here's the insight: If st/u is an integer, then (st/u)2 must also be an integer (squaring an integer always gives an integer). So s2t2/u2 is an integer. Multiply by 3, and 3s2t2/u2 must be an integer. This must be true.
(C) su/t — With our example: s=2, t=3, u=6 gives 12/3 = 4. But try s=1, t=6, u=3. Then st/u = 6/3 = 2 ✓, but su/t = 3/6 = 0.5 ✗. Not always an integer.
(D) tu/s — Try s=2, t=3, u=6. Then tu/s = 18/2 = 9. But try s=3, t=4, u=2. Then st/u = 12/2 = 6 ✓, and tu/s = 8/3 ✗. Not always an integer.
(E) s/tu — This breaks down st/u even further, making it less likely to be an integer.
Answer: B
Common trap: Many students pick (A) thinking "well, if two of the variables divided work out, multiplying all three should too." But the constraint st/u = integer doesn't restrict what happens when you multiply by u again.
Takeaway: When you see "must be true" questions with algebraic expressions, look for ways to directly manipulate the given condition. Here, squaring st/u gave us the structure we needed for option B.
The key concept here is recognizing what "must be true" versus what "could be true." We're told that st/u is an integer — let's call it n. So st/u = n, which means st = nu.
Step 1: Understand what we know
Given: st/u = integer
Step 2: Evaluate each answer choice
Let's test with concrete numbers. Say s=2, t=3, u=6. Then st/u = 6/6 = 1 ✓ (integer)
(A) stu = 2×3×6 = 36. But wait — try s=1, t=2, u=2. Then st/u = 2/2 = 1 ✓, but we can't say stu must be an integer based on our given condition alone.
(B) 3s2t2/u2 — Here's the insight: If st/u is an integer, then (st/u)2 must also be an integer (squaring an integer always gives an integer). So s2t2/u2 is an integer. Multiply by 3, and 3s2t2/u2 must be an integer. This must be true.
(C) su/t — With our example: s=2, t=3, u=6 gives 12/3 = 4. But try s=1, t=6, u=3. Then st/u = 6/3 = 2 ✓, but su/t = 3/6 = 0.5 ✗. Not always an integer.
(D) tu/s — Try s=2, t=3, u=6. Then tu/s = 18/2 = 9. But try s=3, t=4, u=2. Then st/u = 12/2 = 6 ✓, and tu/s = 8/3 ✗. Not always an integer.
(E) s/tu — This breaks down st/u even further, making it less likely to be an integer.
Answer: B
Common trap: Many students pick (A) thinking "well, if two of the variables divided work out, multiplying all three should too." But the constraint st/u = integer doesn't restrict what happens when you multiply by u again.
Takeaway: When you see "must be true" questions with algebraic expressions, look for ways to directly manipulate the given condition. Here, squaring st/u gave us the structure we needed for option B.
