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kevincan
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What is the smallest positive even number 27 Feb 2026, 03:05
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

95% (hard)

Question Stats:

24% (01:29) correct 76% (01:26) wrong based on 38 sessions

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What is the smallest positive even number n such that √(48n) is a multiple of 10?

A. 12
B. 75
C. 150
D. 300
E. 1200
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D
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What is the smallest positive even number 27 Feb 2026, 04:51
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Ans is b
smallest perfect square u need 16*3*3*25, n=3*25=75
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kevincan
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What is the smallest positive even number 27 Feb 2026, 05:36
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Watch out for the word even!
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vasu1104
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What is the smallest positive even number 27 Feb 2026, 07:16
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We need to find smallest even positive integer!
Now 48= 4^2 *3
Now when we take root of it we get 4×(√3n)
Now we can use the given choice to find the ans.

B can be eliminated easily.
For 12 its not possible because 12=3*4^2
Now after simplifying we get 8* 3 and this is not multiple of 10. Reject.

150- 2* 3 * 5^2. Not possible to take the root of it. Reject.

300 - 3 * 2^2* 5^2
So after simplifying we get 4*3*2*5 =120
This definitely is multiple of 10 and also the smallest number.
Ans D
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What is the smallest positive even number 27 Feb 2026, 10:38
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The key concept here is perfect squares and divisibility — and that one word, "even", completely changes the answer.

Key setup: We need sqrt(48n) to be a multiple of 10, meaning 48n must be a perfect square AND divisible by 100.

Step 1 — Express the condition algebraically

sqrt(48n) = 10k for some positive integer k
→ 48n = 100k2
→ n = 25k2/12

For n to be a positive integer, 12 must divide 25k2. Since gcd(25, 12) = 1, we need 12 | k2. Since 12 = 4 x 3: 4|k2 requires 2|k, and 3|k2 requires 3|k. So k must be a multiple of lcm(2,3) = 6.

Step 2 — Find the smallest valid n

Smallest k = 6: n = 25 x 36 / 12 = 75. But 75 is odd — the problem asks for the smallest even n. This is the trap!

Next: k = 12: n = 25 x 144 / 12 = 300. 300 is even. ✓

Verify: sqrt(48 x 300) = sqrt(14400) = 120. Multiple of 10? Yes. ✓

Answer: D. 300

---

Common trap: The instinct is to stop at 75 since it's the first n that makes sqrt(48n) a perfect multiple of 10 — and it does. But 75 is odd. The word "even" in the stem is deliberately easy to miss under exam pressure, which is why 83% of attempted sessions got this wrong.

Takeaway: On "smallest n" problems, list all constraints explicitly before computing — divisibility, parity, and structure conditions each must be verified independently before selecting an answer.

(Kavya | 725 on GMAT Focus Edition)
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