The sum of two positive integers is 52 and their LCM is 168. What is the ratio between the numbers?
Given: x + y = 52 and LCM(x, y) = 168.
Let g be the Greatest Common Divisor (GCD) of x and y. So, x = ga and y = gb, where a and b are coprime positive integers (gcd(a, b) = 1).
Using the given information:
1. x + y = ga + gb = g(a + b) = 52.
2. LCM(x, y) = g * a * b = 168.
From gab = 168, we know that g must be a divisor of 168.
So, g must be a common divisor of 52 and 168.
Divisors of 52: 1, 2, 4, 13, 26, 52
Divisors of 168: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168
Common divisors of 52 and 168 are 1, 2, 4. So g can be 1, 2, or 4.
We also have the constraints: a and b are positive integers and gcd(a, b) = 1.
Case 1: g = 1
a + b = 52/1 = 52
ab = 168/1 = 168
We look for two coprime numbers a and b such that a+b=52 and ab=168. This is equivalent to solving the quadratic equation t² – 52t + 168 = 0. The discriminant is 52² – 4*168 = 2704 – 672 = 2032, which is not a perfect square, so integer solutions for t (a, b) do not exist. Alternatively, list factor pairs of 168 and check sums: (1, 168) sum 169; (2, 84) sum 86; (3, 56) sum 59; (4, 42) sum 46; (6, 28) sum 34; (7, 24) sum 31; (8, 21) sum 29; (12, 14) sum 26. None sum to 52.
Case 2: g = 2
a + b = 52/2 = 26
ab = 168/2 = 84
We look for two coprime numbers a and b such that a+b=26 and ab=84. Factors of 84: (1, 84) sum 85, gcd 1 (valid); (2, 42) sum 44, gcd 2 (invalid); (3, 28) sum 31, gcd 1 (valid); (4, 21) sum 25, gcd 1 (valid); (6, 14) sum 20, gcd 2 (invalid); (7, 12) sum 19, gcd 1 (valid). None sum to 26.
Case 3: g = 4
a + b = 52/4 = 13
ab = 168/4 = 42
We look for two coprime numbers a and b such that a+b=13 and ab=42. Factors of 42: (1, 42) sum 43, gcd 1 (valid); (2, 21) sum 23, gcd 1 (valid); (3, 14) sum 17, gcd 1 (valid); (6, 7) sum 13, gcd 1 (valid).
The pair (6, 7) satisfies a+b=13 and gcd(6, 7)=1.
So, possible values for (a, b) are (6, 7) or (7, 6).
If (a, b) = (6, 7), then x = g*a = 4*6 = 24 and y = g*b = 4*7 = 28.
Check: 24 + 28 = 52. LCM(24, 28) = LCM(2³*3, 2²*7) = 2³*3*7 = 8*21 = 168. This is correct.
The two numbers are 24 and 28.
The ratio between the numbers is 24:28 or 28:24.
24:28 simplifies to 6:7 (dividing by 4).
28:24 simplifies to 7:6 (dividing by 4).
Option C is 7:6.
Therefore, the ratio between the numbers is 7:6 or 6:7. Option C provides 7:6.