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The sum of two positive integers is 156 and their highest common facto

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Bunuel
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The sum of two positive integers is 156 and their highest common facto [#permalink] New post   02 Jul 2020, 00:26
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The sum of two positive integers is 156 and their highest common factor is 13. How many pairs of such numbers are possible?

A. 1
B. 2
C. 3
D. 4
E. 5



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The sum of two positive integers is 156 and their highest common facto [#permalink] New post   02 Jul 2020, 00:43
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Bunuel wrote:
The sum of two positive integers is 156 and their highest common factor is 13. How many pairs of such numbers are possible?

A. 1
B. 2
C. 3
D. 4
E. 5



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First number = 13a
Second number = 13b

where and b are co-prime (relatively Prime) i.e. HCF of a and b = 1



13a+13b = 156
i.e. a+b = 12

i.e (a,b) = (1, 11) (2, 10) (3, 9) (4, 8) (5, 7) (6, 6)

i.e. only two such pairs

Answer: Option B
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The sum of two positive integers is 156 and their highest common facto [#permalink] New post   Updated on: 02 Jul 2020, 00:51
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65 = 5*13
91 = 7*13

143= 13*11
13 = 1*13

HCF in both cases = 13
Sum : 65+91 = 143+13 = 156
Only 2 pairs are possible

Option : B

Originally posted by chipsy on 02 Jul 2020, 00:34.
Last edited by chipsy on 02 Jul 2020, 00:51, edited 1 time in total.
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Re: The sum of two positive integers is 156 and their highest common facto [#permalink] New post   02 Jul 2020, 00:49
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65 = 5*13
91 = 7*13

HCF = 13
Sum : 65+91 = 156
Only 1 pair is possible

Option : A
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Re: The sum of two positive integers is 156 and their highest common facto [#permalink] New post   02 Jul 2020, 00:53
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Nipungupta9081 wrote:
65 = 5*13
91 = 7*13

HCF = 13
Sum : 65+91 = 156
Only 1 pair is possible

Option : A


Copying someone else's explanation is not a very good practice.

I will encourage you to write your own explanation which will improve your understanding, accuracy and content richness of the GC community. :)
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Re: The sum of two positive integers is 156 and their highest common facto [#permalink] New post   02 Jul 2020, 01:12
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Two numbers have the common factor 13, which means that there is no other common factor between the two numbers other than 13.
So let the two numbers be 13A, 13B

\(13(A+B) = 156\)
\(A+B = 12\)

Now, as explained above A and B will be co-primes(have no common factor other than 1)
So, the possible values for A and B are (1,11) and (5,7).

The number of pairs possible = 2

OA, B
Bunuel wrote:
The sum of two positive integers is 156 and their highest common factor is 13. How many pairs of such numbers are possible?

A. 1
B. 2
C. 3
D. 4
E. 5



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Re: The sum of two positive integers is 156 and their highest common facto [#permalink] New post   02 Jul 2020, 02:24
Explanation:
As , 13 is the highest common factor of two number and sum of 2 is 156,
Therefore, 13x+13y = 156
13(x+y) = 156 ------ 1
x+y = 12 ( where x & y has to be co-primes)

Now we have to check values for x & y.
if x = 1, then y = 11 ; put in 1, so sum is 156.
If x= 5 , then y = 7 ; put in 1, so sum is 156.

So only 2 pairs are possible.

IMO-B
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Re: The sum of two positive integers is 156 and their highest common facto [#permalink] New post   26 Jul 2020, 19:09
Bunuel wrote:
The sum of two positive integers is 156 and their highest common factor is 13. How many pairs of such numbers are possible?

A. 1
B. 2
C. 3
D. 4
E. 5


156=13*12

13+13*11= hcf(13)
13*2+13*10=hcf(26)
13*3+13*9=hcf(39)
13*4+13*8=hcf(52)
13*5+13*7=hcf(13)
13*6+13*6=hcf(78)

Only those bold written has 13 hcf

Therefore only 2 pairs possible
Answer B

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Re: The sum of two positive integers is 156 and their highest common facto [#permalink] New post   10 Apr 2023, 02:13
Given: The sum of two positive integers is 156 and their highest common factor is 13.
Asked: How many pairs of such numbers are possible?

Let the two positive integers be x & y.

x + y = 156
HCF(x,y) = 13

x = 13k
y = 13m
where k & m are co-prime.

x + y = 13(k+m) = 156
k + m = 12

(k,m) = {(1,11),(5,7)}

Only 2 pairs (13, 143) & (65,91) are possible.

IMO B
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Re: The sum of two positive integers is 156 and their highest common facto [#permalink]
10 Apr 2023, 02:13
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