[quote]LiveFromThe781 wrote:
red04 wrote:
You could also just know that 7 is a factor of 112/280/175… but that would be too easy
yeah, cause its so obvious.[/quote]
Hey Live, there’s actually a way to always find a common factor between a group of numbers. You do what’s called prime factorization.
So a prime number is a number (like 2, 3, 5, 7, 11, 13, and NOT like 4 = 2 * 2 or 6 = 2 * 3) that doesn’t have any factors other than 1 and itself. Factoring a number into powers of primes goes like this:
Example: 112. Well, 2 definitely goes into 112, because the number ends in 2. So 112 = 2*56. Well, 2 also is a factor of 56, so 56 = 2 * 28. Thus, 112 = 2 * 2 * 28. But 28 = 7 * 4, so that means that 112 = 2 * 2 * 2 * 7 * 4, but this can also be written as 112 = 2^5 * 7.
Example: 280. Since 280 ends in 0, that means that 10 is a factor of 280, 280 = 28 * 10. But 28 = 7 * 4, and 10 = 2 * 5, so 280 = 7 * 4 * 2 * 5 = 7 * 2^3 * 5.
We can do the same thing to find that 175 = 7 * 25 = 7 * 5^2. So, 7 is a factor of all three numbers, because it is in their prime factorization!
So, 112 x^2 + 280 x + 175 = 7 (32 x^2 + 40 x + 25), and that is a lot easier to factor.
Key tips:
Do it a step at a time. You’ll get better with practice, but try first dividing out by small primes if you don’t immediately see a bigger one that works. For example, try 2, 3, and 5 if all else fails.
There are tricks to tell if a number goes into another number evenly.
- 2 is a factor of any number that ends in 0, 2, 4, 6, or 8 (even numbers) and is NOT a factor of any odd number.
- 5 goes into any number ending in 0 or 5.
- 10 is a factor of all numbers ending in 0.
- (This one’s a little tricky) 3 is a factor of a number only if it goes evenly into the sum of its digits. For example, 12345 has a factor of 3, because 1 + 2 + 3 + 4 + 5 = 15 = 3 * 5. The same rule works for 9.
Give these a try. But honestly, if you can practice the quadratic formula, that is often even better!