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The sum of two positive integers is 52 and their LCM is 168. What is the ratio between the numbers ?
Explanation
Let the two positive integers be x and y. Let their Highest Common Factor (HCF) be h. We can define x = ha and y = hb, where a and b are co-prime integers.
Given the following conditions:
1. Sum: h(a + b) = 52
2. LCM: h × a × b = 168
Dividing the equations to eliminate h:
h(a + b)⁄hab = 52⁄168
a + b⁄ab = 13⁄42
We need two co-prime numbers whose sum is 13 and product is 42. Solving the quadratic equation t2 - 13t + 42 = 0 gives t = 6 and t = 7. Thus, a = 6 and b = 7. Substituting back into the sum equation, h(6 + 7) = 52, so h = 4.
The numbers are 4 × 6 = 24 and 4 × 7 = 28. The ratio between them is 28:24 (or 24:28), which simplifies to 7:6.
SIMILAR QUESTIONS
If the LCM and HCF of two positive integers are 18 and 3 respectively, then what is the minimum possible value of their sum ?
The sum of two numbers is 168. If their HCF is 14, how many such pairs of numbers are possible?
When the square of the sum of two numbers are added to the square of their difference, we get 416. The difference between the square of the sum and square of the difference is 384. What are the numbers ?