Question

(a) find the largest four digit number which has a total of exactly 3 factors assuming that one and d number itself are factors. (b) find a and b greater than zero such that a^2 = b^3. (c) find x and y such that x^3 y^3 =1729

Answer 1

sounds like you want a perfect square. Any other kind of number will have at least 4 factors: 1,n,q, and n/q or be a prime, with only 2 factors.

So, what's the largest square n <= 9999?
√9999 = 99.99, so
99^2 = 9801 has just 3 factors: 1,99,9801


64 = 8^2 = 4^3

1729 = 1^3 + 12^3 = 9^3 + 10^3
It is the smallest number which can be expressed as the sum of two cubes in two different ways.

Answer 2

b)8^2=4^3 (c) 1729=12^3+1^3=10^3+9^3