# confusion on factoring a number into it's prime factors

I just started working through Algebra Demystified and there's a section for learning how to factor a number into it's prime factors. The description says...

To factor a number into its prime factors (those which have no divisors other than themselves and 1), start with a list of prime numbers. Begin with the smallest prime number and keep dividing the prime numbers into the number to be factored. Stop dividing when the square of the prime number is larger than the number.

In one of the examples given they ask you to give the prime factors of 166. The solution they provide is 2, 7 and 11. I don't understand how they got these numbers. I googled factoring a number into prime factors and kept seeing factor trees which for 166 would give me 2 and 83. Using the book instructions should give me 2, 3, 5, 7, and 11 if I'm understanding it correctly.

Shouldn't the factors go evenly into the final number? If so, that would be 2 and 83.

And how does squaring the prime factor cause you to stop? If the prime factors of 166 are 2 and 83, I don't understand how squaring works into the equation. Thanks.

• Typos and arithmetic errors are always possible, but I find it hard to believe that the text said that $166$ is divisible by $7$ or $11$. In any case, it is not. – lulu Mar 31 '18 at 19:46
• I apologize, the book says "The list of prime numbers to check are 2, 7 and 11" Now that I reread that I see they're not saying they are a factor, just prime numbers to check. But not sure why 3 and 5 aren't in the list. – geoff swartz Mar 31 '18 at 19:47
• Well, $3$ and $5$ are trivially checked. As is $2$, of course, so I don't know why they'd list $2$ but not the others. – lulu Mar 31 '18 at 19:49