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Let me know if you need more explanation
Date Posted:
4 years ago
thanks!but why largest possible value of N is 131?
Recall that the 10th multiple of 12 is 120 and the 11th multiple of 12 is 132.
Once you hit the next number 132, there will be a total of 11 numbers from 1 to 132 which are divisible by both 3 and 4 (ie by 12). However, the question provides the information that n (A n B) = 10, meaning there are only 10 numbers in this set (ie the first 10 multiples of 12, up to 120).
Since n (A n B) is not 11, we cannot include the 11th multiple of 12 (ie 132). This means that N cannot be 132 and above, because one of the elements of A n B would automatically be 132 (since this number is now found in both the epsilon set and the A n B set) and this would automatically change n (A n B) from 10 to 11.
Increasing the value of N beyond 120 is still possible as long as N does not get to 132, since there is no multiple of 12 between 120 and 132 (and hence, this will not add to the n (A n B) count from 10 to 11). For example, N can be 125, because the multiples of 12 from 1 to 125 are 12, 24, ..., 108, 120. Having the epsilon go to 125 does not affect the number of numbers divisible by 12.
Because of this reasoning, N can go as large as 131.
Once you hit the next number 132, there will be a total of 11 numbers from 1 to 132 which are divisible by both 3 and 4 (ie by 12). However, the question provides the information that n (A n B) = 10, meaning there are only 10 numbers in this set (ie the first 10 multiples of 12, up to 120).
Since n (A n B) is not 11, we cannot include the 11th multiple of 12 (ie 132). This means that N cannot be 132 and above, because one of the elements of A n B would automatically be 132 (since this number is now found in both the epsilon set and the A n B set) and this would automatically change n (A n B) from 10 to 11.
Increasing the value of N beyond 120 is still possible as long as N does not get to 132, since there is no multiple of 12 between 120 and 132 (and hence, this will not add to the n (A n B) count from 10 to 11). For example, N can be 125, because the multiples of 12 from 1 to 125 are 12, 24, ..., 108, 120. Having the epsilon go to 125 does not affect the number of numbers divisible by 12.
Because of this reasoning, N can go as large as 131.
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Hope this helps, impt parts in pink
Date Posted:
4 years ago