Two positive integers multiply to 5070, with their highest common factor being 13 and their sum amounting to 143. What is the difference between these two integers?
Let the two numbers be ‘13a’ and ‘13b’ ATQ; 13a × 13b = 5070 So, (a × b) = 30 ……………………. (1) Also, 13a + 13b = 143 So, a + b = 11 Or, a = (11 – b) …………………... (2) Putting value of ‘a’ in equation (1), we have (11 – b) × b = 30 Or, 11b – b2 = 30 Or, b2 – 11b + 30 = 0 Or, b2 – 6b – 5b + 30 = 0 Or, b(b – 6) – 5(b – 6) = 0 Or, (b – 6)(b – 5) = 0 So, b = 6 or b = 5 So, the two numbers are = 13 × 5 = 65 and 13 × 6 = 78 Required difference = 78 - 65 = 13
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