Prime Factorization

Step-by-step solution

1. 1

   Start

   Factorize 840 by trial division: test candidate divisors in increasing order (2, 3, 5, 7, 11, …) and divide whenever one fits, until the remainder itself is prime.

2. 2

   Divide by 2 (3 times)

   840 is divisible by 2 more than once: 840 ÷ 2 = 420, 420 ÷ 2 = 210, 210 ÷ 2 = 105. The prime 2 appears 3 times.

   $$840 = 2 \cdot 420 = 2^{2} \cdot 210 = 2^{3} \cdot 105$$
3. 3

   Divide by 3

   105 is divisible by 3: 105 ÷ 3 = 35.

   $$105 = 3 \cdot 35$$
4. 4

   Divide by 5

   35 is divisible by 5: 35 ÷ 5 = 7.

   $$35 = 5 \cdot 7$$
5. 5

   Divide by 7

   Every candidate divisor up to √7 has been tested, so the remaining number 7 is itself prime — the last factor: 7 ÷ 7 = 1.

   $$7 = 7 \cdot 1$$
6. 6

   Prime factorization

   Collect all the prime factors found along the way: 840 = 2^3 · 3 · 5 · 7.

   $$840 = 2^{3} \cdot 3 \cdot 5 \cdot 7$$