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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If a is positive integer, can a be expressed as the product of two int

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Math Expert V
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If a is positive integer, can a be expressed as the product of two int  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 55% (01:17) correct 45% (01:27) wrong based on 105 sessions

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If a is positive integer, can a be expressed as the product of two integers, each of which is greater than 1?

(1) a is even.
(2) 7 < a < 11

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If a is positive integer, can a be expressed as the product of two int  [#permalink]

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1
1
i) since a is even - it can only be a +ve integer.
if a = 2 = 2*1 but x,y have to be >1 therefore not A

ii) a is a +ve integer
let a = 9 =3*3, but x,y have to be >1 (false)
let a = 8 = 4*2 or 8*1 (false)

B

Originally posted by MG1788 on 26 Dec 2016, 23:12.
Last edited by MG1788 on 27 Dec 2016, 01:51, edited 2 times in total.
Manager  G
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Re: If a is positive integer, can a be expressed as the product of two int  [#permalink]

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OhsostudiousMJ wrote:
Bunuel wrote:
If a is positive integer, can a be expressed as the product of two integers, each of which is greater than 1?

(1) a is even.
(2) 7 < a < 11

Hi, the question asks if a "can" be expressed as.....

And in both the cases, of course, a can be expressed as asked.

Well the first statement is not valid when a = 2. Ideally we say a statement is true when it is valid for all the values, so here in this case say Can 2 be expressed as product of 2 ints both greater than 1? The answer is no , hence s1 is false.
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Re: If a is positive integer, can a be expressed as the product of two int  [#permalink]

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MG1788 wrote:
i) since a is even - it can only be a +ve integer.
if a = 2 = 2*1 but x,y have to be >1 therefore not A

ii) a can be a decimal or integer
let a = 8.1 = 0.3*27 but x,y have to be >1
let a = 10.1 = 10.1*1 but x,y have to be >1
therefore not B

i) + ii)
a = even and lies between 7 & 11 =? a = 8,10. these can be represented as x*y with x,y>1

I think itc C

I agree till last line , but i think it is E

a = 8 or 10 : 8 = 4 * 2 here both are >1

8 = 8*1 here one integer is equal to 1
Marshall & McDonough Moderator D
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Re: If a is positive integer, can a be expressed as the product of two int  [#permalink]

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Question: Is 'a' composite integer?

If 'a' is prime it cannot be expressed as a product of 2 integers, each of which is greater than 1.

St1: a is even --> If a = 2 then a is prime and cannot be expressed as a product of 2 integers which are greater than 1.
If a = 4 then 4 = 2*2 and can be expressed as product of 2 integers.
Not Sufficient.

St2: a = 8, 9, 10 --> a is not prime. a can be expressed as product of 2 integers which are greater than 1.
Sufficient.

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If a is positive integer, can a be expressed as the product of two int  [#permalink]

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IMO E

Given - a >0, a->integer
asked can a = b*c with b,c>1 and b,c integers

1) since a is even
Possible solutions
(i) b=2, c=1 a=even
a = 2*1 = 2 ( c not greater than 1 ) ---------NO
(ii) b=2, c=3 a=even
a = 2*3 = 6 -------------- YES

2) a = { 8,9,10 }
Possible Solutions
(i) b = 5 c = 2
a= 5*2 = 10 ----------- YES ( B,C integers, >1 )
(ii) b= 5/2 c=4
a = 5/2 * 4 =10 -------------- NO ( B,C >1 but B is not an integer. But A still meets the criteria )

Therefore, Not A, not B

Combine (1) and (2) statements

Possible Solutions
(i) b = 5 c = 2
a= 5*2 = 10 ----------- YES ( B,C integers, >1 )
(ii) b= 5/2 c=4
a = 5/2 * 4 =10 -------------- NO ( B,C >1 but B is not an integer)

A is even and a lies {8,9,10}
But we still have YES and NO answers.

Hence E.
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Re: If a is positive integer, can a be expressed as the product of two int  [#permalink]

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jojo1 wrote:
IMO E

Given - a >0, a->integer
asked can a = b*c with b,c>1 and b,c integers

1) since a is even
Possible solutions
(i) b=2, c=1 a=even
a = 2*1 = 2 ( c not greater than 1 ) ---------NO
(ii) b=2, c=3 a=even
a = 2*3 = 6 -------------- YES

2) a = { 8,9,10 }
Possible Solutions
(i) b = 5 c = 2
a= 5*2 = 10 ----------- YES ( B,C integers, >1 )
(ii) b= 5/2 c=4
a = 5/2 * 4 =10 -------------- NO ( B,C >1 but B is not an integer. But A still meets the criteria )

Therefore, Not A, not B

Combine (1) and (2) statements

Possible Solutions
(i) b = 5 c = 2
a= 5*2 = 10 ----------- YES ( B,C integers, >1 )
(ii) b= 5/2 c=4
a = 5/2 * 4 =10 -------------- NO ( B,C >1 but B is not an integer)

A is even and a lies {8,9,10}
But we still have YES and NO answers.

Hence E.

Hi jojo1,

Question is not asking us to test whether a or b is a non integer. We have to check whether the integer can be expressed as a product of two integers greater than 1.
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If a is positive integer, can a be expressed as the product of two int  [#permalink]

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Bunuel wrote:
If a is positive integer, can a be expressed as the product of two integers, each of which is greater than 1?

(1) a is even.
(2) 7 < a < 11

Hi, the question asks if a "can" be expressed as.....

And in both the cases, of course, a can be expressed as asked.

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Re: If a is positive integer, can a be expressed as the product of two int  [#permalink]

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OhsostudiousMJ wrote:
Bunuel wrote:
If a is positive integer, can a be expressed as the product of two integers, each of which is greater than 1?

(1) a is even.
(2) 7 < a < 11

Hi, the question asks if a "can" be expressed as.....

And in both the cases, of course, a can be expressed as asked.

Well the first statement is not valid when a = 2. Ideally we say a statement is true when it is valid for all the values, so here in this case say Can 2 be expressed as product of 2 ints both greater than 1? The answer is no , hence s1 is false.

Okay, I get it now. Thanks.
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Re: If a is positive integer, can a be expressed as the product of two int  [#permalink]

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I think the answer is B.

Statement 1 tells us that A has to be even, but an even integer allows for both scenarios:

Case 1: 2*3 = 6 = Product of two integers that are not 1
Case 2: 1*2 = 2 = Not possible to be formed by multiplying two positive integers that are not 1

Statement two tells us that a is not prime (no primes between 7 and 11)
Therefore, we know that "a" can be formed by multiplying two integers other than one.
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Re: If a is positive integer, can a be expressed as the product of two int  [#permalink]

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Given a = + ve integer
Que = a can be expressed as c *b or not

1. A is even ,lets start with 2 = 2* no feasible value as value should be > 1
with 4 = 2 * 2 = feasible
NOT SUFFICIENT
2. 7 < a < 11
a = 2*4 , 2*5 , 4 * 2 , 5 *2 feasible
SUFFICIENT'

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Re: If a is positive integer, can a be expressed as the product of two int  [#permalink]

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Given :
a=mn, where m & n are greater than 1 (m&n>1)
from st1 -> a is even i.e a = 2,4,6,8,10.....
lets take it one by one :
a = 2 ----> mn = 2 ----> 2*1 = 2 (not valid --> condition is m & n >1)
a = 4 ----> mn = 4 ----> 2*4 = 2 (valid)

from these solutions, we can see there is no consistency, so NOT SUFFICIENT

from st2 -> 7 < a < 11, which means values of a are {8,9,10}
lets take it one by one :
8 = possible factors = 1,2,4,8 = (1*8);(2*4);(4*2);(8*1) = 1 being factor of 8, makes it against the cond
& so not valid
9 = possible factors = 1,3,9 = (1*9);(3*3);(9*1) = 1 being factor of 9, makes it against the cond & so
not valid
10 = possible factors = 1,2,5,10 = (1*10);(2*5);(5*2);(10*1) = 1 being factor of 10, makes it against the cond & so not valid

so a ={8,9,10} results a STRICTLY NO

Note : In DS, either STRICTLY YES or STRICTLY NO results a SUFFIECIENT RESULT

Therefore B is Suffient. Re: If a is positive integer, can a be expressed as the product of two int   [#permalink] 15 Jan 2020, 01:39
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