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Factors and Prime Factors
Every positive, non-prime integer has at least two prime factors.
21 is a product of two prime numbers, 3 and 7
38 is a product of two prime numbers, 2 and 19
If a question asks for the ‘prime factorization’ of 38, the answer is “ 2 and 19” If the question is ‘how many prime factors
does 38 have’, the answer is ‘two’
If a question asks about the factors of 38, the answer is ‘four’ positive integer factors . Any integer, whether prime or not, has
at least two positive integer factors, namely the number 1 and the number itself.
Now, let us kick this up a notch. Take the integer 12. What is its prime factors? The answer is ‘two’ – namely 2 and 3.
12 can be written as (23)(3)
On the other hand, if the question is about [positive integer] ‘factors’ then we maust include the following: 1, 2, 3, 4, 6, 12
In other words, the integer 12 has six positive integer factors. Remember, when you deoineate them, always to ‘bookend’ the
list with the number ‘1’ and the factored number itself.
Next level
How many [positive integer] factors does 84 have?
Applying the basic rule we just learned, we start with 1 and 84, and then all we have got to do, is fill in all the numbers inbetween that are factors of 84. A smart way to do that, is to go up the ladder – ask yourself is 84 divisible by 2 if yes, then add
2 and 42 to the list. Next, ask if 84 is divisible by 3. If yes, then add 3 and 84/3 = 28 to the list
In summary, we obtain the following list as the factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
Let us see if there is an easy way to hone in on his. 84 can be written as (2)(2)(3)(7)
If you treat this like a combination problem, you can show that the number of distinct possible SINGLE factors = 3
The number of distinct possible factors involving the mere product of TWO of its inherent factors,, is (4C2)/2 + 1=4
The number of distinct possible factors involving the mere product of THREE of its inherent factors,, is (4C3)=3
The number of distinct possible factors involving the mere product of FOUR of its inherent factors,, is (4C4) = 1
These add up to 11 – but we must also add the number ‘1’ – as you know, 1 is not a prime number, so we could not include it
in the above analysis. Another way to say this, is that we also obtain a factor of 84 by utilizing zero of its prime factors –
namely the factor ‘1’
In summary, the integer 84 has 12 positive integer factors.
Wouldn’t it be neat if there were a FORMULA for all of this – it sure would save us a lot of time and heartache on the GMAT.
You are in luck – there is:
All you have to do, is commit the formula to memory
Remember that 84 can be written as (22)(31)(71) - here we give you the formula:
If Q consists of (P1)x(P2)y(P3)z then Q has (x +1)(y +1)(z +1) factors (where P1 …P3 are distinct prime
factors of Q)
Under this formula, 84 would have to have (2 + 1)(1 + 1)(1 +1) factors. That is to say, 3 times 2 times 2 = 12
Therefore, if the GMAT question asks you to figure out how many factor the integer 1001 has – not to worry. Apply the
formula.
The integer 1001 is often found on the GMAT because it is the product of 7 times 11 times 13.
Hence, 1001 would have (1 + 1) (1 + 1) (1 + 1) factors, or 8 factors.
They would be: 1, 7, 11, 13, 7⋅⋅11, 11⋅⋅13, 7⋅⋅13, 7⋅⋅11⋅⋅13