How many pairs of positive integers x, y exist such that HCF of x, y = 35 and…

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How many pairs of positive integers x, y exist such that HCF of x, y = 35 and sum of x and y = 1085 ?

  1. A.

    30

  2. B.

    15

  3. C.

    12

  4. D.

    8

Attempted by 20 students.

Show answer & explanation

Correct answer: B

Step-by-Step Solution

To find the number of pairs of positive integers (x, y) given their HCF and sum, follow these steps:

  1. Use the HCF property: If HCF(x, y) = 35, we can express the numbers as: x = 35a y = 35b where a and b are coprime integers (meaning HCF(a, b) = 1).

  2. Use the sum property: x + y = 1085 35a + 35b = 1085 Dividing by 35: a + b = 1085 / 35 = 31

  3. Find the number of coprime pairs (a, b) that sum to 31: We need pairs (a, b) such that a + b = 31 and HCF(a, b) = 1. Since 31 is a prime number, any pair (a, b) where a + b = 31 will naturally be coprime, provided a and b are positive. The possible pairs (a, b) are: (1, 30), (2, 29), (3, 28), (4, 27), (5, 26), (6, 25), (7, 24), (8, 23), (9, 22), (10, 21), (11, 20), (12, 19), (13, 18), (14, 17), (15, 16).

    Counting these, we get exactly 15 pairs.

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