If arithmetic and geometric mean of two positive numbers are 5 and 3, then…

For two positive numbers, if the arithmetic mean (AM) and geometric mean (GM) are known, the numbers are the two roots of the quadratic equation formed from their sum and product: sum = 2 × AM and product = GM2.

1. Arithmetic mean = 5, so x + y = 2 × 5 = 10.

2. Geometric mean = 3, so xy = 32 = 9.

3. x and y are roots of t2 − 10t + 9 = 0 (since sum = 10, product = 9).

4. Factoring: (t − 9)(t − 1) = 0, so t = 9 or t = 1.

Cross-check: (9 + 1) / 2 = 5 (matches the given AM) and √(9 × 1) = √9 = 3 (matches the given GM).

Hence, the required numbers are 9 and 1.