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28 Mar 2026, 00:18
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
63% (02:39) correct
38%
(02:19)
wrong
based on 32
sessions
History
Date
Time
Result
Not Attempted Yet
How many positive integers less than or equal to 300 are divisible by both 3 and 4 but not by 8 or 9?
A. 5
B. 6
C. 7
D. 8
E. 9
A. 5
B. 6
C. 7
D. 8
E. 9
29 Apr 2026, 00:06
Kudos
Bookmarks
How many positive integers less than or equal to 300 are divisible by both 3 and 4 but not by 8 or 9?
We need to find all such positive numbers that are less than or equal to 300, divisible by 3 and 4, but not by 8 or 9:
Let's first find out the count of all such numbers that are less than or equal to 300, divisible by 3 and 4:
A number divisible by both 3 and 4 must be divisible by the LCM of 3 & 4 = LCM(3, 4) = 12
Count of multiples of 12 that are less than or equal to 300 = Greatest integer of (300/12) = 25
Hence, there are 25 multiples of 12.
Now, we need to check for the numbers that are divisible by 8 in these 25 multiples:
Multiples of 12 that are also divisible by 8 => LCM(12, 8) = 24
Count of multiples of 24 that are less than or equal to 300 = Greatest integer of (300/24) = 12
Remaining numbers = 25 - 12 = 13
Next, we need to check for the numbers that are divisible by 9 in these 25 multiples:
Multiples of 12 that are also divisible by 9 => LCM(12, 9) = 36
Count of multiples of 36 that are less than or equal to 300 = Greatest integer of (300/36) = 8
Remaining numbers = 13 - 8 = 5
Adding back the overlapping numbers (Inclusion-Exclusion Principle):
Some numbers are divisible by: 12, 8, and 9 simultaneously => LCM(12, 8, 9) = 72
Multiples of 72 that are less than or equal to 300 => Greatest integer of (300/72) = 4
These 4 numbers were:
removed once when subtracting multiples of 24 (÷8)
removed again when subtracting multiples of 36 (÷9)
So, we need to account for them in our final calculation.
Thus, final answer = 5+4 = 9
Hence, the correct answer is Option E - 9.
Hope this helps!
We need to find all such positive numbers that are less than or equal to 300, divisible by 3 and 4, but not by 8 or 9:
Let's first find out the count of all such numbers that are less than or equal to 300, divisible by 3 and 4:
A number divisible by both 3 and 4 must be divisible by the LCM of 3 & 4 = LCM(3, 4) = 12
Count of multiples of 12 that are less than or equal to 300 = Greatest integer of (300/12) = 25
Hence, there are 25 multiples of 12.
Now, we need to check for the numbers that are divisible by 8 in these 25 multiples:
Multiples of 12 that are also divisible by 8 => LCM(12, 8) = 24
Count of multiples of 24 that are less than or equal to 300 = Greatest integer of (300/24) = 12
Remaining numbers = 25 - 12 = 13
Next, we need to check for the numbers that are divisible by 9 in these 25 multiples:
Multiples of 12 that are also divisible by 9 => LCM(12, 9) = 36
Count of multiples of 36 that are less than or equal to 300 = Greatest integer of (300/36) = 8
Remaining numbers = 13 - 8 = 5
Adding back the overlapping numbers (Inclusion-Exclusion Principle):
Some numbers are divisible by: 12, 8, and 9 simultaneously => LCM(12, 8, 9) = 72
Multiples of 72 that are less than or equal to 300 => Greatest integer of (300/72) = 4
These 4 numbers were:
removed once when subtracting multiples of 24 (÷8)
removed again when subtracting multiples of 36 (÷9)
So, we need to account for them in our final calculation.
Thus, final answer = 5+4 = 9
Hence, the correct answer is Option E - 9.
Hope this helps!
