Find the least natural number that is divisible by three different primes.

Question

31-10-2026
Mathematics

Answered

Find the least natural number that is divisible by three different primes.

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-10-31T11:23:06-04:00

Identify the sequence of prime numbers as $2, 3, 5, 7, 11, \dots$.
Select the three smallest distinct prime numbers to minimize the final product: $2, 3, 5$.
Multiply these primes together to find the least natural number divisible by all three: $2 \times 3 \times 5 = 30$.
Conclude that the least natural number is $\boxed{30}$.

Explanation

Problem Analysis and Identification of Smallest Primes

A natural number is a positive integer, such as $1, 2, 3, \dots$. We are looking for the least natural number that is divisible by three different prime numbers. A prime number is a natural number greater than $1$ that has no positive divisors other than $1$ and itself. The sequence of the smallest prime numbers is $2, 3, 5, 7, 11, \dots$. To minimize a product of distinct factors, we must choose the smallest available factors. Therefore, to find the least natural number divisible by three distinct prime numbers, we choose the three smallest prime numbers: $2$, $3$, and $5$.

Calculating the Least Natural Number

Since the number must be divisible by $2$, $3$, and $5$, and we want the smallest such number, we calculate the least common multiple of these three prime numbers. Because $2$, $3$, and $5$ are distinct prime numbers, their least common multiple is simply their product: $2 \times 3 \times 5$.

Final Verification and Result

Performing the multiplication: $2 \times 3 = 6$, and $6 \times 5 = 30$. Thus, the number $30$ is the smallest natural number divisible by three different primes ($2$, $3$, and $5$). Any other combination of three distinct primes would involve larger numbers (for example, $2 \times 3 \times 7 = 42$ or $2 \times 5 \times 7 = 70$), resulting in a larger product.

Final Answer

The least natural number that is divisible by three different primes is $\boxed{30}$.

Examples

Understanding the properties of prime numbers and their products is foundational for fields like cryptography. For example, in RSA encryption, large numbers are formed by multiplying two very large distinct primes to ensure security. Smaller versions of this logic are used in computer science for resource scheduling or setting up repeating cycles in machinery, where you need to find the smallest common interval that accounts for different frequency requirements. Finding the least number divisible by specific prime constraints helps in optimizing these systems for efficiency.