How many Prime Factors does 111 111 have?

111,111 has 5 distinct prime factors.

Understanding the prime factors of a number involves breaking it down into its smallest prime constituents. A prime factor is a prime number that divides the given number exactly.

Step-by-Step Prime Factorization of 111,111

To determine the number of prime factors for 111,111, we first need to find its prime factorization.

Breaking Down 111,111

The number 111,111 can be expressed as a product of two smaller, more manageable numbers:
111,111 = 111 × 1001

Factorizing 111

The number 111 is relatively small and can be easily factored. It is known that:

• 111 is divisible by 3 (since the sum of its digits, 1+1+1=3, is divisible by 3).
• 111 ÷ 3 = 37
• 37 is a prime number.

Therefore, the prime factorization of 111 is:
111 = 3 × 37

Factorizing 1001

Now, we need to find the prime factors of 1001:

• Divisibility by 7: Let's check if 1001 is divisible by 7.
  • 1001 ÷ 7 = 143
• Factorizing 143: We now need to find the prime factors of 143.
  • 143 is not divisible by 2, 3, 5, or 7.
  • Let's try the next prime number, 11.
  • 143 ÷ 11 = 13
  • 13 is a prime number.

Therefore, the prime factorization of 1001 is:
1001 = 7 × 11 × 13

Combining All Factors

By combining the prime factors of 111 and 1001, we get the complete prime factorization of 111,111:

111,111 = 111 × 1001
111,111 = (3 × 37) × (7 × 11 × 13)

Arranging the prime factors in ascending order, the distinct prime factors of 111,111 are: 3, 7, 11, 13, and 37.

Summary of Prime Factors

The prime factorization reveals the distinct prime numbers that compose 111,111.

Counting these distinct prime factors, we find there are 5 unique prime numbers. Each of these factors appears only once in the prime factorization.

In conclusion, the number 111,111 has 5 distinct prime factors.