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Math Expert V
Joined: 02 Sep 2009
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Does positive integer n have positive integer factors a and b such tha  [#permalink]

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Difficulty:   65% (hard)

Question Stats: 67% (02:06) correct 33% (02:16) wrong based on 35 sessions

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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

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Math Expert V
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Re: Does positive integer n have positive integer factors a and b such tha  [#permalink]

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Does positive integer n have positive integer factors a and b such that ab = n and a - b = 1?

a-b=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..

(2) n = 19,073
Answer is NO as n is odd but has to be even

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Re: Does positive integer n have positive integer factors a and b such tha  [#permalink]

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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 (a-b=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  [#permalink]

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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

a-b=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; 3-2=1
n=12; ab=4,3; 4-3=1

(1) When n is divided by 10, the remainder is 3 sufic

n/10 r=3: n={3,13,23…103…}=odd, so the answer to the question is no.

(2) n = 19,073 sufic

n=odd, so the answer to the question is no.

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Re: Does positive integer n have positive integer factors a and b such tha  [#permalink]

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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

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Re: Does positive integer n have positive integer factors a and b such tha  [#permalink]

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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.

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Re: Does positive integer n have positive integer factors a and b such tha  [#permalink]

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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

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Re: Does positive integer n have positive integer factors a and b such tha  [#permalink]

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We are to determine if the positive integer factors such that ab=n and a-b=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.

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Re: Does positive integer n have positive integer factors a and b such tha  [#permalink]

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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   [#permalink] 02 Dec 2019, 21:45
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