I am the 8th highest composite number. What am I?

[zackattack]

I am the 8th highest composite number. What am I?

I am a triangular number. The sum of my digits is 6. I am divisible by 5. What am I?

I am one of a pair of identical prime factors of 60. What am I?

I am a 3 digit number below 300 and divisible by 10. My digits add up to 6. I have 4 factors. What am I?

[stapel_eliz]

I am the 8th highest composite number. What am I?

A composite number is a non-prime, a number which is the product of two numbers not equal to 1. So list the first few composites, and count up to the eighth one.

I am a triangular number. The sum of my digits is 6. I am divisible by 5. What am I?

"Triangular" numbers are one which can be pictures as triangles, such as:

* *
 *

* * *
 * *
  *

* * * *
 * * *
  * *
   *

So start listing the numbers which are multiples of 5. Find one whose digits add up to 6 (such as 60, 150, etc). Draw the pictures until you find one that can be drawn as a triangle of dots.

I am one of a pair of identical prime factors of 60. What am I?

Find the prime factorization of 60. Look for two copies of the same factor.

I am a 3 digit number below 300 and divisible by 10. My digits add up to 6. I have 4 factors. What am I?

So the number is between 100 and 299. Being divisible by 10, it is between 100 and 290.

The digits add up to 6, so the digits have to be a combination of 6 and 0, 5 and 1 and 0, 4 and 2 and 0, or 3 and 3 and 0. (Anything else wouldn't give you a sum of six with only three digits.)

You can't use the 6 and 0 option, because the only possible number is 600, which is too big. For 5, 1, and 0, you can only use 150, since 510 is too big and 501 and 105 are not divisible by 10. And so forth.

Find the (few) numbers which will work, and then factorize. The one with four factors is the one you want.