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Is the positive integer n a multiple of 24 ?
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26 Feb 2014, 01:33
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siddhans wrote:
If in the same question we were to replace 'multiple' by 'divisible' what the difference??? What exactly happens when something is a multiple of something or when something is a divisbile of something ?
There is no difference between saying that 12 is a multiple of 4 and that 12 is divisible by 4.
As for the question. Is the positive integer n a multiple of 24?
(1) n is a multiple of 4. Not sufficient. (2) n is a multiple of 6. Not sufficient.
(1)+(2) n is a multiple of both 4 and 6 which means that it's a multiple of least common multiple of 4 and 6, which is 12. So, even taken together statements are not sufficient, since n can be for example 12 as well as 24. Not sufficient.
Answer: E.
Generally if a positive integer n is a multiple of positive integer a and positive integer b, then n is a multiple of LCM(a,b).
Yes, you are right. I ignored the fact that the 2 in the prime factor of 6 may be the same 2 from the prime factor of 2's in the factors of 12. Thus, n definitely has only two 2's and one 3 as factor, which is 12. thanks.
Yes, you are right. I ignored the fact that the 2 in the prime factor of 6 may be the same 2 from the prime factor of 2's in the factors of 12. Thus, n definitely has only two 2's and one 3 as factor, which is 12. thanks.
fluke please make clear in your way such as:
Prime factors of 24: 2^3*3 (1) 4: 2^2; Not sufficient. (2) 6: 2*3; Not sufficient.
Combining both; minimum factors of n= 2^2*2*3 = 2^3*3 = all factors of 24. Sufficient. _________________
Yes, you are right. I ignored the fact that the 2 in the prime factor of 6 may be the same 2 from the prime factor of 2's in the factors of 12. Thus, n definitely has only two 2's and one 3 as factor, which is 12. thanks.
fluke please make clear in your way such as:
Prime factors of 24: 2^3*3 (1) 4: 2^2; Not sufficient. (2) 6: 2*3; Not sufficient.
Combining both; minimum factors of n= 2^2*2*3 = 2^3*3 = all factors of 24. Sufficient.
If in the same question we were to replace 'multiple' by 'divisible' what the difference??? What exactly happens when something is a multiple of something or when something is a divisbile of something ?
Is the positive integer n a multiple of 24 ?
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Updated on: 17 May 2021, 08:46
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Bunuel wrote:
Is the positive integer n a multiple of 24 ?
(1) n is a multiple of 4. (2) n is a multiple of 6.
Target question:Is the positive integer n a multiple of 24 ? This is a good candidate for rephrasing the target question. -----ASIDE--------------------- A lot of integer property questions can be solved using prime factorization. For questions involving divisibility, divisors, factors and multiples, we can say: If N is a multiple of k, then k is "hiding" within the prime factorization of N
Consider these examples: 24 is a multiple of 3 because 24 = (2)(2)(2)(3) Likewise, 70 is a multiple of 5 because 70 = (2)(5)(7) And 112 is a multiple of 8 because 112 = (2)(2)(2)(2)(7) And 630 is a multiple of 15 because 630 = (2)(3)(3)(5)(7) -----ONTO THE QUESTION!---------------------
Since 24 = (2)(2)(2)(3), we can rephrase the target question as follows... REPHRASED target question:Are three 2's and one 3 hiding in the prime factorization of n?
Aside: the video below has tips on rephrasing the target question
Statement 1: n is a multiple of 4. 4 = (2)(2), so statement 1 is telling us that there are two 2's hiding in the prime factorization of n. Of course, there COULD be additional 2's (and 3's for that matter) hiding in the prime factorization of n. Given this, there's no way we can answer the target question with certainty. If you're not convinced, consider these two possible cases: Case a: n = 12 (which is a multiple of 4). In this case, the answer to the target question is NO, n is not divisible by 24 Case b: n = 48 (which is a multiple of 4). In this case, the answer to the target question is YES, n is divisible by 24 Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: n is a multiple of 6 6 = (2)(3), so statement 2 is telling us that there is one 2 and one 3 hiding in the prime factorization of n. Of course, there COULD be additional 2's and 3's for that matter hiding in the prime factorization of n. Given this, there's no way we can answer the target question with certainty. Consider these two possible cases: Case a: n = 12 (which is a multiple of 6). In this case, the answer to the target question is NO, n is not divisible by 24 Case b: n = 48 (which is a multiple of 6). In this case, the answer to the target question is YES, n is divisible by 24 Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED. In other words, Case a: n = 12 (which is a multiple of 4 and 6). In this case, the answer to the target question is NO, n is not divisible by 24 Case b: n = 48 (which is a multiple of 4 and 6). In this case, the answer to the target question is YES, n is divisible by 24 Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT
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