234, 519 and 78 prime method
Answers
Answer:
1) Here are the first several prime factors: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...
Let's start by dividing 234 by 2
234 ÷ 2 = 117 - No remainder! 2 is one of the factors!
117 ÷ 2 = 58.5 - There is a remainder. We can't divide by 2 evenly anymore. Let's try the next prime number
117 ÷ 3 = 39 - No remainder! 3 is one of the factors!
39 ÷ 3 = 13 - No remainder! 3 is one of the factors!
13 ÷ 3 = 4.3333 - There is a remainder. We can't divide by 3 evenly anymore. Let's try the next prime number
13 ÷ 5 = 2.6 - This has a remainder. 5 is not a factor.
13 ÷ 7 = 1.8571 - This has a remainder. 7 is not a factor.
13 ÷ 11 = 1.1818 - This has a remainder. 11 is not a factor.
13 ÷ 13 = 1 - No remainder! 13 is one of the factors!
The orange divisor(s) above are the prime factors of the number 234. If we put all of it together we have the factors 2 x 3 x 3 x 13 = 234. It can also be written in exponential form as 21 x 32 x 131.
2) 519 = 167 + 173 + 179, three consecutive primes. The difference between each of those prime numbers is 6 so one of them is also one of the factors of 519. The other factor that pairs up with it should be obvious, too. 519 is the hypotenuse of the Pythagorean triple 156-495-519.
3) 78 = 1 x 78, 2 x 39, 3 x 26, or 6 x 13. Factors of 78: 1, 2, 3, 6, 13, 26, 39, 78. Prime factorization: 78 = 2 x 3 x 13.