Sum of two numbers x, y = 1050. What is the maximum value of the HCF between x…

Key insight: the greatest common divisor (HCF) must divide the sum 1050.

Let d be the HCF of the two positive numbers. Then we can write the numbers as d·a and d·b for positive integers a and b. Therefore d(a+b)=1050.

• To maximize d, we must minimize (a+b). The smallest possible value of (a+b) for positive integers a and b is 2 (when a=1 and b=1).

• With (a+b)=2 we get d = 1050/2 = 525, which is achieved by the numbers 525 and 525.

• If the problem required the two numbers to be distinct, the minimal (a+b) is 3 (for example a=1, b=2), giving d = 1050/3 = 350, achieved by 350 and 700.

Therefore the maximum possible HCF is 525 (and 350 if the two numbers must be distinct).