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Does positive integer n have positive integer factors a and b such tha
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02 Dec 2019, 00:16
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Competition Mode Question Does positive integer n have positive integer factors a and b such that ab = n and a  b = 1? (1) When n is divided by 10, the remainder is 3 (2) n = 19,073 Are You Up For the Challenge: 700 Level Questions
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Re: Does positive integer n have positive integer factors a and b such tha
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02 Dec 2019, 04:28
Does positive integer n have positive integer factors a and b such that ab = n and a  b = 1? ab=1..a=b+1, so ab=b(b+1). Thus the question finally become whether n is a product of TWO consecutive integers.(1) When n is divided by 10, the remainder is 3 So n=10a+3, and 10a+3 will always be ODD, but product of consecutive integers is always even.. So, answer is NO (2) n = 19,073 Answer is NO as n is odd but has to be even D
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Re: Does positive integer n have positive integer factors a and b such tha
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02 Dec 2019, 05:55
The answer is D. Quote: Does positive integer n have positive integer factors a and b such that ab = n and a  b = 1? The question stem actually asks us whether n is a product of two consecutive integers (ab=1 >> a=b+1). The product of two consecutives will be always EVEN, so if we are provided with information that n is ODD, we will know for sure that the answer to the question is NO.
Quote: (1) When n is divided by 10, the remainder is 3  it means that unit digit of n is 3, the fact that implies that n is ODD, so it cannot be a product of two consecutive, so the reply is NO. Sufficient. Quote: (2) n = 19,073 Same logic as during analysis of statement A: n is odd, so the reply is NO. Sufficient.



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Re: Does positive integer n have positive integer factors a and b such tha
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02 Dec 2019, 06:06
Quote: Does positive integer n have positive integer factors a and b such that ab = n and a  b = 1?
(1) When n is divided by 10, the remainder is 3
(2) n = 19,073 ab=1 means they are consecutive integers, thus one must be even and the other odd. If ab=even*odd=even, then n=even. n=6; ab=3,2; 32=1 n=12; ab=4,3; 43=1 (1) When n is divided by 10, the remainder is 3 suficn/10 r=3: n={3,13,23…103…}=odd, so the answer to the question is no. (2) n = 19,073 suficn=odd, so the answer to the question is no. Ans (D)



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Re: Does positive integer n have positive integer factors a and b such tha
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02 Dec 2019, 08:53
Since a=b+1, n=ab can be rewritten as n=(b+1)b. Since b and (b+1) are two consecutive positive integers, n has to be a positive even integer.
So, the real question is whether n is a positive even integer or not.
(1) n=3,13,23,... So, any value of n is always a positive odd integer. This precisely answers the question. SUFFICIENT
(2) n = 19,073... So n is obviously a positive odd integer. This precisely answers the question. SUFFICIENT
Final answer is (D)
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Re: Does positive integer n have positive integer factors a and b such tha
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02 Dec 2019, 11:34
Does positive integer n have positive integer factors a and b such that ab = n and a  b = 1?
(1) When n is divided by 10, the remainder is 3
(2) n = 19,073
(1) n = (2x * 5y) + 3; where x and y are integers. When n = 13, although ab (1*13) = n, but a b >1. Again when n = 63, which can be prime factorized as 3^2 * 7, ab = n ,but a  b > 1. So, n can be expressed as a product of an integer with unit digit of 1 and 3, the difference between its 2 factors will always be greater than 1. Sufficient.
(2) n =19,703 = 17* 19*61. none of them will result in a  b = 1. sufficient.
D is the answer IMHO.



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Re: Does positive integer n have positive integer factors a and b such tha
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02 Dec 2019, 18:25
n— positive integer a, b — positive integers
ab= n a —b = 1 —> a and b — consecutive integers
In order “ a,b” to be positive integer factors of n, —> n must be even numbers Is a an even???
(Statement1): When n is divided by 10, the remainder is 3 n is odd number (Always no) Sufficient
(Statement2) n = 19,073 n is odd number —> (Always No) Sufficient
The answer is D
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Re: Does positive integer n have positive integer factors a and b such tha
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02 Dec 2019, 19:06
We are to determine if the positive integer factors such that ab=n and ab=1. From this question, we know that a and b are two consecutive integers. By extension, we also know that the condition can only be satisfied if n is even. With this idea in mind, let's evaluate the statements.
Statement 1: When n is divided by 10, the remainder is 3 From statement 1, we know that n is odd. Since we know that a basic requirement of n as stated above is that n must at least be even, we can conclude that two factors, a and b do not exist for n such that a and b are consecutive integers. Statement 1 is sufficient.
Statement 2: n=19073 Once again, we can conclude that n can never be a product of two consecutive integers since n is odd. Statement 2 is also sufficient.
The answer is D.



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Re: Does positive integer n have positive integer factors a and b such tha
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02 Dec 2019, 21:45
Does positive integer n have positive integer factors a and b such that ab = n and a  b = 1?
(1) When n is divided by 10, the remainder is 3.......satisfied by 3,13,23,33,43........... so we can say that n wont be having consecutive factors.....sufficient
(2) n = 19,073......which is a prime no with factors 1 and itself........so we can say that n wont be having consecutive factors .......sufficient
OA:D




Re: Does positive integer n have positive integer factors a and b such tha
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02 Dec 2019, 21:45






