Number properties HCF question.

1. Number Theory

2. Number Theory
The sum of two non co–prime numbers added to their HCF gives us
91. How many such pairs are possible?
(a) 2 (b) 4
(c) 3 (d) 6

3. Number Theory
Let HCF of the numbers be h. The numbers can be taken as ha + hb,
where a, b are coprime.
h + ha + hb = 91
h(1 + a + b) = 91
h  1
h = 7 => 1 + a + b = 13 a + b = 12
h = 13 => 1 + a + b = 7
a + b = 6
The sum of two non co–prime numbers added to their HCF gives us
91. How many such pairs are possible?

4. Number Theory
Case 1: h = 7, a + b = 12
(1, 11), (5, 7) => Only 2 pairs are possible as a, b have to be coprime.
Case 2: h = 13, a + b = 6
(1, 5) only one pair is possible as a, b have to be coprime.
Overall, 3 pairs of numbers are possible – (7, 77) (35, 49) and (13, 65)
Answer choice (c)
The sum of two non co–prime numbers added to their HCF gives us
91. How many such pairs are possible?

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