what is the least natural number that has exactly five distinct, positive factors?
To find the least natural number that has exactly five distinct, positive factors, we need to understand what a factor is. A factor of a number is a whole number that divides that number evenly without leaving a remainder.
Let's break this problem down step-by-step:
Step 1: The number must have exactly five distinct, positive factors. Distinct factors mean that no factor is repeated, and positive factors mean that the factors are greater than zero.
Step 2: We need to determine how many factors a number can have. The number of factors a natural number has can be calculated by finding the prime factorization of the number and using the formula: (a + 1)(b + 1)(c + 1)... where a, b, c, etc., are the exponents of the prime factors.
Step 3: Given that the number must have exactly five distinct, positive factors, we know that we need three prime factors raised to the power of 4, 2, and 0, respectively, to achieve a total of five factors.
Step 4: Now, let's find the least number that satisfies these conditions. Start by choosing the smallest prime numbers, which are 2, 3, and 5, and raise them to the powers mentioned above.
2^4 * 3^2 * 5^0 = 16 * 9 * 1 = 144
Thus, the least natural number that has exactly five distinct, positive factors is 144.