what is the least positive integer with exactly 10 factors?
To find the least positive integer with exactly 10 factors, we need to understand how factors of a number work.
A factor is a number that divides another number evenly. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 without leaving a remainder.
The number of factors a positive integer has depends on its prime factorization. If a number is expressed as a product of prime factors raised to various powers, the total number of factors can be calculated using these powers.
Now, let's find the least positive integer with exactly 10 factors. To do this, we need to consider the different possibilities for prime factorization.
The prime factorization of 10 can be written as 2^1 * 5^1. Therefore, a number with exactly 10 factors must have two prime factors: one raised to the power of 1 and the other raised to the power of 1.
To find the least positive integer with these properties, we need to consider the smallest prime numbers, which are 2 and 3. However, 3 cannot be one of the prime factors since it is not part of the prime factorization of 10.
Therefore, we are left with 2 and 5 as possible prime factors. To form the least positive integer, we multiply these prime factors together, giving us the number 2 * 5 = 10.
Hence, the least positive integer with exactly 10 factors is 10.