find the smallest positive integer that should be added to 1500 to make it a perfect square
LectureNotes asked, “Find the smallest positive integer that should be added to 1500 to make it a perfect square?”
Answer:
To find the smallest positive integer that needs to be added to 1500 to make it a perfect square, let’s first express 1500 as the product of its prime factors:
1500 = 2^2 \times 3 \times 5^3
To make it a perfect square, we need to balance the exponents of each prime factor. For that, let’s identify which prime factors are missing or have insufficient power.
- 2 Factorization: We already have 2^2, so it’s balanced.
- 3 Factorization: We only have one (3), so we need an additional (3) to make it (3^2).
- 5 Factorization: We have 5^3 which is already more than enough to make it a perfect square.
To make (1500) a perfect square, we need to add another (3) to make it (3^2), as the least common multiple of all prime factors will give us the smallest integer to be added.
So, the smallest positive integer that should be added to (1500) to make it a perfect square is (3).
After adding (3) to (1500), we get (1503), which is (3^2 \times 167), a perfect square.