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How many pairs of positive integers whose highest common factor is 12 13 Jan 2021, 10:00
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B
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How many pairs of positive integers whose highest common factor is 12 add up to 216?

A. 2
B. 3
C. 9
D. 17
E. 18
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B
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manasi05
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How many pairs of positive integers whose highest common factor is 12 Updated on: 13 Jan 2021, 13:42
Originally posted by manasi05 on 13 Jan 2021, 13:35.
Last edited by manasi05 on 13 Jan 2021, 13:42, edited 2 times in total.
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Sum of pairs of positive numbers with gcf 12 sums up to 216
So for some value of x and y 12x+12y = 216
so x+y = 18
So x y needs to be in pair of sum 18. From these pairs we have to find the numbers whose gcf is 12 only
12*1 + 12*17 = 216 (12 is the gcf)
12*2 + 12*16 = 216 (gcf is 24)
This way we will get 3 pairs where 12 will be the gcf
(1,17), (5,13), (7,11)
So 3 such pairs
IMO answer will be B
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KrishKapoor
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How many pairs of positive integers whose highest common factor is 12 13 Jan 2021, 13:45
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Can you explain what this question means

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How many pairs of positive integers whose highest common factor is 12 Updated on: 14 Jan 2021, 08:23
Originally posted by manasi05 on 13 Jan 2021, 13:57.
Last edited by manasi05 on 14 Jan 2021, 08:23, edited 1 time in total.
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KrishKapoor
Can you explain what this question means

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Sure.
The question mentions that the sum of pairs of integers whose greatest common factor is 12 equals 216
So those pairs will be multiple of 12 as the gcf is 12
So if suppose we have 12 and 204 as integers, their gcf will be 12 and sum will be 216
We have to find such pairs
So if we have integers x and y then we know that x and y will be multiple of 12
From that, we can form an equation
12x+12y = 216
x+y = 18
So x and y will be pairs whose sum will be equal to 18 such as (1,17), (2,16), (3,15) and so on
Now lets substitute those values of x and y pair in equation
12*1 + 12*17 = 216 ---- (Here we have only 12 as gcf between these 2 numbers as 12 can only be taken as common)
12*2 + 12*16 = 216 ---- (Here we can take 12*2 as gcf as that can be taken as common term between both numbers)
In a similar way check for other pairs and you will see only 3 such pairs exist where you can not take anything common other than 12 so their GCF is 12.

If you didn't understand any point then do let me know.
Thanks.
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How many pairs of positive integers whose highest common factor is 12 14 Jan 2021, 10:48
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Bunuel
How many pairs of positive integers whose highest common factor is 12 add up to 216?

A. 2
B. 3
C. 9
D. 17
E. 18
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This means that the two number have greatest factor as 12. The two number, therefore, can be 12a and 12b, where a and b are co-prime, that is a and b have no common factors.
Thus 12a+12b=216.......a+b=18
So either both a and b are odd or both are even.
I) The moment both are even, they are not co-prime as 2 becomes a common factor.
II) Therefore, both a and b are odd. They can be
1+17
3+15(NO, as 3 is common factor)
5+13
7+11
9+9(NO, as 9 is common)

So 3 pairs - (1,17);(5,14);(7,11)

B
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How many pairs of positive integers whose highest common factor is 12 30 Mar 2022, 22:25
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Bunuel
How many pairs of positive integers whose highest common factor is 12 add up to 216?

A. 2
B. 3
C. 9
D. 17
E. 18

Answer:

Let,Two pairs of positive integer be x & y.
Since the Common factor is 12 & their sum is 216
So, 12x+12y=216
Or, x+y=18
So, the pairs of integer sum must be 18 & their common factor is 1.
We get the pairs (1,17),(5,13),(7,11) which common factor is 1.
So answer is 3 pairs

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How many pairs of positive integers whose highest common factor is 12 10 Sep 2025, 10:54
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