(x, y) is a pair of positive integers such that LCM (x, y) - HCF (x, y) = 143 and x > y.

Which one of the following is correct in respect of the Question and the Statements given below?

Statement 1: There are three possible values of HCF (x, y).

Statement 2: The minimum value of (x - y) is 7.

Statement 3: There are 6 pairs of (x, y) that satisfy the given conditions.

Correct option is D

​Solution:

​Let d = HCF(x, y)

Then, x = $$d \times x_1$$​ and y = $$d \times y_1$$​, where $$\gcd(x_1, y_1) = 1$$​ and x_1 > y_1

The LCM is:

LCM(x, y) = $$\frac{x \times y}{\text{HCF}(x, y)} = \frac{(d \times x_1) \times (d \times y_1)}{d} = d \times x_1 \times y_1.$$​​

Given: LCM(x, y) - HCF(x, y) = 143.

Substituting:

​$$d \times x_1 \times y_1 - d = 143$$​​

Factorizing:

​$$d(x_1 y_1 - 1) = 143$$​​

Factorize 143 and Finding d, $$x_1 y_1$$​ Pairs

143 = $$11 \times 13$$​​

Possible pairs: $$(d, x_1 y_1 - 1)$$​ = (1, 143), (11, 13), (13, 11), (143, 1)

Thus: $$d = 1, x_1 y_1$$​= 144

​$$d = 11, \quad x_1 y_1 = 14 \\d = 13, \quad x_1 y_1 = 12 \\d = 143, \quad x_1 y_1 = 2$$​​

Finding Valid (x, y) Pairs

Case 1: $$d = 1, x_1 y_1 = 144$$​​

Factors of 144: (144,1), (72,2), (48,3), (36,4), (24,6), (18,8), (16,9), (12,12)

Valid coprime pairs: (144,1), (16,9)

Pairs: (144,1), (16,9)

Case 2: $$d = 11, x_1 y_1 = 14$$​​

Factors of 14: (14,1), (7,2)

Valid coprime pairs: (14,1), (7,2)

Pairs: (154,11), (77,22)

Case 3: $$d = 13, x_1 y_1 = 12$$​​

Factors of 12: (12,1), (6,2), (4,3)

Valid coprime pairs: (12,1), (4,3)

Pairs: (156,13), (52,39)

Case 4: $$d = 143, x_1 y_1 = 2$$​​

Factors of 2: (2,1)

Valid coprime pair: (2,1)

Pair: (286,143)

Found pairs: (144,1), (16,9), (154,11), (77,22), (156,13), (52,39), (286,143)

Total: 7 pairs

Verifying the Statements

Statement 1: There are three possible values of HCF(x,y)

HCF values are 1, 11, 13, 143.}

Excluding trivial d = 143,

the HCF values are 1, 11, 13.

Thus, Statement 1 is Correct.

Statement 2: The minimum value of (x − y) is 7.

Compute x−y:

144 - 1 = 143

16 - 9 = 7

154 - 11 = 143

77 - 22 = 55

156 - 13 = 143

52 - 39 = 13

286 - 143 = 143

Minimum difference is 7. Thus, Statement 2 is Correct.

Statement 3: There are 6 pairs of (x,y) that satisfy the given conditions. Excluding (286,143), there are exactly 6 pairs.

Thus, Statement 3 is Correct

Thus, all three statements are correct.