Trial and error is the way, but there are certain "streamlining" methods.

1) Rules of divisibility can tell you whether a number is divisible by a prime, without going all the way and trying to divide it:

A number is divisible by 2 if it ends with an even digit. Example: 342.
A number is divisible by 3 if the sum of its digits is divisible by 3. Example: 225: 2+2+5=9 is divisible by 3, so 225 is divisible by 3.
A number is divisible by 4 if last two digits form a number that is div by 4: Example: 344: 44 is div by 4, so 344 is divisible by 4.
A number is divisible by 5 if it ends with an a 5 or zero. Example: 345.
A number is divisible by 6 if it's divisible by 2 and 3 - see above.
A number is divisible by 7...if it's divisible by 7. Use know multiples of 7 to check this: your number 133 is 7 below 140, which is a known multiple of 7, so 133 is a multiple of 7.
A number is divisible by 8 if last three digits form a number that is div by 8: Example: 1720: 720 is div by 8, so 1720 (or 2720, or 23720) are all divisible by 8.
A number is divisible by 9 if the sum of its digits is divisible by 9. Example: 225: 2+2+5=9 is divisible by 9, so 225 is divisible by 9.
A number is divisible by 10 if it ends with an a zero. Example: 340.

So you can use the rules to just test whether there is any need to actually try and divide the number by a prime, before you go and actually do it (for example, before going out and realizing that 133/7 = 19.

2) When testing whether a number is prime or divisible by primes, it's enough to stop testing when you reach the nearest perfect square. For 133, the nearest perfect square is 121, which is the square of 11: therefore, you need to test only until 11 to see whether 133 is prime. If 133 is not divisible by any prime up to 11, inclusive, then 133 is prime itself, and there's no need to test any further. Try it with 131, or 149.

Hey lawri,

Let me just add on to Geva's informative post by adding one more point:

One of the quickest ways to factor a large number is to look at the answer choices. After all, if your goal in factorization is to reduce your answer, the correct answer must be something that you could derive from what you have.

For example, if you're at 124/93 and the answer choices are:

A) 6/5
B) 5/4
C) 4/3
D) 3/2
E) 2/3

You don't necessarily need to know right offhand that both 124 and 93 are divisible by 31. You can just let the answer choices guide you:

A doesn't work because 93 cannot be broken in to 5-times-something, so A is out
B doesn't work because the 124 isn't divisible by 5, and 93 isn't divisible by 4
C looks viable, so you can hold onto it
D doesn't work because 93 isn't divisible by 2
E may still work, but look again - the numerator is less than the denominator, which isn't true of 124/93, so that doesn't work either.


The GMAT won't typically require too much brute force or trial and error calculation - they'll almost always give you an out if you can leverage the information. Often times those answer choices are a great tool to help you with that, especially when it comes to factoring fractions or roots.

Geva@MasterGMAT wrote:Trial and error is the way, but there are certain "streamlining" methods.

1) Rules of divisibility can tell you whether a number is divisible by a prime, without going all the way and trying to divide it:

A number is divisible by 2 if it ends with an even digit. Example: 342.
A number is divisible by 3 if the sum of its digits is divisible by 3. Example: 225: 2+2+5=9 is divisible by 3, so 225 is divisible by 3.
A number is divisible by 4 if last two digits form a number that is div by 4: Example: 344: 44 is div by 4, so 344 is divisible by 4.
A number is divisible by 5 if it ends with an a 5 or zero. Example: 345.
A number is divisible by 6 if it's divisible by 2 and 3 - see above.
A number is divisible by 7...if it's divisible by 7. Use know multiples of 7 to check this: your number 133 is 7 below 140, which is a known multiple of 7, so 133 is a multiple of 7.
A number is divisible by 8 if last three digits form a number that is div by 8: Example: 1720: 720 is div by 8, so 1720 (or 2720, or 23720) are all divisible by 8.
A number is divisible by 9 if the sum of its digits is divisible by 9. Example: 225: 2+2+5=9 is divisible by 9, so 225 is divisible by 9.
A number is divisible by 10 if it ends with an a zero. Example: 340.

So you can use the rules to just test whether there is any need to actually try and divide the number by a prime, before you go and actually do it (for example, before going out and realizing that 133/7 = 19.

2) When testing whether a number is prime or divisible by primes, it's enough to stop testing when you reach the nearest perfect square. For 133, the nearest perfect square is 121, which is the square of 11: therefore, you need to test only until 11 to see whether 133 is prime. If 133 is not divisible by any prime up to 11, inclusive, then 133 is prime itself, and there's no need to test any further. Try it with 131, or 149.

Geva,

Thanks for the response and listing out these rules. I especially like the idea of using a know multiple to check / determine if a number is divisible by seven. I suppose this method should work just as well for other numbers.

Brian@VeritasPrep wrote:Hey lawri,

Let me just add on to Geva's informative post by adding one more point:

One of the quickest ways to factor a large number is to look at the answer choices. After all, if your goal in factorization is to reduce your answer, the correct answer must be something that you could derive from what you have.

For example, if you're at 124/93 and the answer choices are:

A) 6/5
B) 5/4
C) 4/3
D) 3/2
E) 2/3

You don't necessarily need to know right offhand that both 124 and 93 are divisible by 31. You can just let the answer choices guide you:

A doesn't work because 93 cannot be broken in to 5-times-something, so A is out
B doesn't work because the 124 isn't divisible by 5, and 93 isn't divisible by 4
C looks viable, so you can hold onto it
D doesn't work because 93 isn't divisible by 2
E may still work, but look again - the numerator is less than the denominator, which isn't true of 124/93, so that doesn't work either.


The GMAT won't typically require too much brute force or trial and error calculation - they'll almost always give you an out if you can leverage the information. Often times those answer choices are a great tool to help you with that, especially when it comes to factoring fractions or roots.

Brian,

Thanks for the input, I like it very much. The challenge I find is to quickly size up a question and logically pick it apart in two minutes. Once I identify a sound strategy it comes together pretty fast.