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Difficulty:
25%
(medium)
Question Stats:
79% (02:02) correct
21%
(01:53)
wrong
based on 43
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If j is a positive integer such that the square of j is divisible by 96 and 98 but not by 99 or 100, j could be a multiple of which of the following ?
A. 10
B. 11
C. 15
D. 16
E. 20
A. 10
B. 11
C. 15
D. 16
E. 20
14 Mar 2026, 23:35
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Let's first prime-factorize all the numbers:
96 = 2^5 * 3
98 = 2 * 7^2
For j^2 to be divisible by both: j^2 must contain 2^5, 3 and 7^2
Now, if j^2 has p^k, then j needs half the power (rounded up).
So, j must contain: 2^3 (because of 2^5), 3, 7
Hence, j must contain (2^3)*3*7 = 168 => j is a multiple of 168.
It is also given that j is not divisible by 99 or 100.
99 = 3^2 * 11
Since j^2 must not contain 3^2, j cannot contain 3^2. So j has exactly one 3.
100 = 2^2 * 5^2
So j^2 must not contain 5^2. Thus j cannot contain 5.
Now, let's check each option, one by one:
Option A - j is a multiple of 10
Contains 5, thus, Option A is incorrect.
Option B - j is a multiple of 11
j^2 has 11^2, but 99 divides 11^2*3^2 and since j already has a 3, this could cause the j^2 to be divisible by 99.
Option C - j is a multiple of 15
Contains 5, thus, Option C is incorrect.
Option D - j is a multiple of 16
16 = 2^4
This should be correct. Example: j = 168 * 2 = 336
Check: j^2 is divisible by 96 and 98, and is not divisible by 99 and 100.
Option E - j is a multiple of 20
Contains 5, thus, Option E is incorrect.
Hence, the correct answer is Option D - j is a multiple of 16.
Hope this helps!
96 = 2^5 * 3
98 = 2 * 7^2
For j^2 to be divisible by both: j^2 must contain 2^5, 3 and 7^2
Now, if j^2 has p^k, then j needs half the power (rounded up).
So, j must contain: 2^3 (because of 2^5), 3, 7
Hence, j must contain (2^3)*3*7 = 168 => j is a multiple of 168.
It is also given that j is not divisible by 99 or 100.
99 = 3^2 * 11
Since j^2 must not contain 3^2, j cannot contain 3^2. So j has exactly one 3.
100 = 2^2 * 5^2
So j^2 must not contain 5^2. Thus j cannot contain 5.
Now, let's check each option, one by one:
Option A - j is a multiple of 10
Contains 5, thus, Option A is incorrect.
Option B - j is a multiple of 11
j^2 has 11^2, but 99 divides 11^2*3^2 and since j already has a 3, this could cause the j^2 to be divisible by 99.
Option C - j is a multiple of 15
Contains 5, thus, Option C is incorrect.
Option D - j is a multiple of 16
16 = 2^4
This should be correct. Example: j = 168 * 2 = 336
Check: j^2 is divisible by 96 and 98, and is not divisible by 99 and 100.
Option E - j is a multiple of 20
Contains 5, thus, Option E is incorrect.
Hence, the correct answer is Option D - j is a multiple of 16.
Hope this helps!
