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09 Mar 2026, 03:50
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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:
95%
(hard)
Question Stats:
41% (02:36) correct
59%
(02:55)
wrong
based on 81
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How many positive integers less than or equal to 900 are multiples of 12 but are not perfect squares?
A. 67
B. 70
C. 71
D. 72
E. 75
A. 67
B. 70
C. 71
D. 72
E. 75
09 Mar 2026, 10:31
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kevincan
How many numbers are multiples of 12, which are less than 900.
12K <= 900
Thus, K <= 75
The value of K lies between 1 to 75. As, positive integer is greater than 0.
The number of possible values of K = 75.
12k = 2^2 * 3 * k
We need to find perfect squares.
K = 3*1 OR
K= 3*4 OR
K = 3 * 5^2
K= 3 * 3^2
K= 3 * 2^4
These 5 values are Perfect Squares.
So, Multiples of 12, but not perfect squares = 75 - 5 = 70.
Option B
09 Mar 2026, 14:45
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The key concept here is Sets and Counting with Overlap — specifically finding elements in one set that are excluded from another. This is a classic application of the Inclusion-Exclusion principle.
Step 1: Find all multiples of 12 up to 900.
Divide 900 by 12: 900 ÷ 12 = 75.
So there are 75 multiples of 12 from 12 to 900.
Step 2: Identify which of those 75 are also perfect squares.
A number that is both a multiple of 12 and a perfect square must be a perfect square that is divisible by 12. For a perfect square to be divisible by 12 = 22 × 3, it must contain at least 22 and 32 in its prime factorization (since perfect squares have only even exponents). So the number must be divisible by 22 × 32 = 36.
This means we need perfect squares that are multiples of 36 — i.e., multiples of 36 that are also perfect squares. These are: 36, 144, 324, 576, 900 — which are (62), (122), (182), (242), (302).
Check: 302 = 900 ✓, 362 = 1296 > 900 ✗.
So there are 5 such numbers.
Step 3: Subtract.
Multiples of 12 that are NOT perfect squares = 75 − 5 = 70.
Answer: B (70)
The common trap: Almost everyone gets 75 (answer E) and moves on — they find the multiples of 12 correctly but forget to remove the ones that are also perfect squares. The question specifically says "but are not perfect squares," and under time pressure that clause gets skipped.
A subtler trap: students who do think to subtract sometimes only check multiples of 12 that are perfect squares of integers (12, 24, 36...) without reasoning through the prime factorization. That leads them to list too many or too few overlapping values.
Takeaway: Whenever a counting question has two conditions joined by "but not," set up both groups first, find the overlap carefully using prime factorization if divisibility is involved, then subtract.
Step 1: Find all multiples of 12 up to 900.
Divide 900 by 12: 900 ÷ 12 = 75.
So there are 75 multiples of 12 from 12 to 900.
Step 2: Identify which of those 75 are also perfect squares.
A number that is both a multiple of 12 and a perfect square must be a perfect square that is divisible by 12. For a perfect square to be divisible by 12 = 22 × 3, it must contain at least 22 and 32 in its prime factorization (since perfect squares have only even exponents). So the number must be divisible by 22 × 32 = 36.
This means we need perfect squares that are multiples of 36 — i.e., multiples of 36 that are also perfect squares. These are: 36, 144, 324, 576, 900 — which are (62), (122), (182), (242), (302).
Check: 302 = 900 ✓, 362 = 1296 > 900 ✗.
So there are 5 such numbers.
Step 3: Subtract.
Multiples of 12 that are NOT perfect squares = 75 − 5 = 70.
Answer: B (70)
The common trap: Almost everyone gets 75 (answer E) and moves on — they find the multiples of 12 correctly but forget to remove the ones that are also perfect squares. The question specifically says "but are not perfect squares," and under time pressure that clause gets skipped.
A subtler trap: students who do think to subtract sometimes only check multiples of 12 that are perfect squares of integers (12, 24, 36...) without reasoning through the prime factorization. That leads them to list too many or too few overlapping values.
Takeaway: Whenever a counting question has two conditions joined by "but not," set up both groups first, find the overlap carefully using prime factorization if divisibility is involved, then subtract.
