If n is an integer, is n/7 an integer?
(1) 3n/7 is an integer.
(2) 5n/7 is an integer.
OA is D
What is the correct way in solving this kind of question ? I am getting E. Below are the two methods
Method 1:
statement 1: 3n/7 = x (int)
Next step, n/7 = x/3 ==> gives both integer and non integer
same goes for statement 2. Hence E
Method 2:
Is n/7 an integer ? Is n/7 = m(int) ==> Is n = 7m ? (Is n a multiple of 7 ?)
Statement 1: 3n/7 = x (int)
Next step, 3n = 7x ==> n = 7x/3 (given n is an integer, so it is sufficient)
Same goes for statement 2. Hence D
Thanks & Regards
Sachin
If n is an integer, is n/7 an integer?
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- sachin_yadav
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Method 2 looks good to me.sachin_yadav wrote:If n is an integer, is n/7 an integer?
(1) 3n/7 is an integer.
(2) 5n/7 is an integer.
OA is D
What is the correct way in solving this kind of question ? I am getting E. Below are the two methods
Method 1:
statement 1: 3n/7 = x (int)
Next step, n/7 = x/3 ==> gives both integer and non integer
same goes for statement 2. Hence E
Method 2:
Is n/7 an integer ? Is n/7 = m(int) ==> Is n = 7m ? (Is n a multiple of 7 ?)
Statement 1: 3n/7 = x (int)
Next step, 3n = 7x ==> n = 7x/3 (given n is an integer, so it is sufficient)
Still I would do it another way. Here it is, barely any math.
Statement 1 says that 3n/7 is an integer. 3 is not a multiple of 7 and 7 is not a multiple of 3. So multiplying an integer by 3 changes nothing related to being a multiple of 7. So if n is an integer and 3n is a multiple of 7, n is a multiple of 7.
Sufficient
Statement 2 does the same thing with 5. 5 is not a multiple of 7 and 7 is not a multiple of 5. So the same logic applies.
Sufficient
Done.
Choose D.
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Statement 1: 3n/7 is an integer.sachin_yadav wrote:If n is an integer, is n/7 an integer?
(1) 3n/7 is an integer.
(2) 5n/7 is an integer.
3n/7 = a, where a is an integer.
Thus:
3n = 7a
n = (7/3)a.
For n to be an integer -- as required by the prompt -- a must be a multiple of 3.
If a=3, then n = (7/3)(3) = 7.
If a=6, then n = (7/3)(6) = 14.
If a=9, then n = (7/3)(9) = 21.
The resulting values for n are all multiples of 7.
Thus, n/7 must be an integer.
SUFFICIENT.
Statement 2: 5n/7 is an integer
5n/7 = b, where b is an integer.
Thus:
5n = 7b
n = (7/5)b.
For n to be an integer -- as required by the prompt -- b must be a multiple of 5.
If b=5, then n = (7/5)(5) = 7.
If b=10, then n = (7/5)(10) = 14.
If b=15, then n = (7/5)(15) = 21.
The resulting values for n are all multiples of 7.
Thus, n/7 must be an integer.
SUFFICIENT.
The correct answer is D.
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A lot of integer property questions can be solved using prime factorization.sachin_yadav wrote:If n is an integer, is n/7 an integer?
(1) 3n/7 is an integer.
(2) 5n/7 is an integer.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N
Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)
--------------------------------
Okay, onto the question:
Target question: Is n/7 an integer?
Statement 1: 3n/7 is an integer
If 3n/7 is an integer, then we can also say that 3n is DIVISIBLE by 7.
This means that there's a 7 HIDING in the prime factorization of 3n.
Since there's no 7 HIDING in 3, there must be a 7 HIDING in the prime factorization of n.
If there's a 7 HIDING in the prime factorization of n, then n must be divisible by 7
If n is divisible by 7, then n/7 is DEFINITELY an integer.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT
Statement 2: 5n/7 is an integer
If 5n/7 is an integer, then we can also say that 5n is DIVISIBLE by 7.
This means that there's a 7 HIDING in the prime factorization of 5n.
Since there's no 7 HIDING in 5, there must be a 7 HIDING in the prime factorization of n.
If there's a 7 HIDING in the prime factorization of n, then n must be divisible by 7
If n is divisible by 7, then n/7 is DEFINITELY an integer.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT
Answer = D
Cheers,
Brent
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I'd recommend that you compare this question with OG #135, which tests a similar principle.
In this problem, your second interpretation is much better: "is n/7 an integer -> is n divisible by 7." It's generally better to approach divisibility problems in conceptual terms rather than algebraic ones.
If in this problem one of the statements had read "14n/7 is an integer," then the statement would be INSUFFICIENT. Since 14 itself is divisible by 7, n could be anything and the statement would still be true.
Since 3 and 5 are not divisible by 7, then if 3n and 5n are each divisible by 7, n must be divisible by 7 - the 3 and 5 can't be helping to divide 7, as 14 was in the previous example.
If in a case like this the coefficient in the numerator shares no factors with the denominator, then the variable must be divisible by the denominator.
In this problem, your second interpretation is much better: "is n/7 an integer -> is n divisible by 7." It's generally better to approach divisibility problems in conceptual terms rather than algebraic ones.
If in this problem one of the statements had read "14n/7 is an integer," then the statement would be INSUFFICIENT. Since 14 itself is divisible by 7, n could be anything and the statement would still be true.
Since 3 and 5 are not divisible by 7, then if 3n and 5n are each divisible by 7, n must be divisible by 7 - the 3 and 5 can't be helping to divide 7, as 14 was in the previous example.
If in a case like this the coefficient in the numerator shares no factors with the denominator, then the variable must be divisible by the denominator.
Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
EdM in Mind, Brain, and Education
Harvard Graduate School of Education