"How many odd factors does

"If n is the smallest integer such that

"How many prime numbers are factors of

If questions like these make you cringe, I'd like to convince you that only a few easy-to-understand concepts stand between you and doing these flawlessly.

This is probably review, but just for a refresher: a prime number is any positive integer that is divisible by only

In preparation for the QUANT, it would be good to be familiar with this list. If you verify for yourself why each number from

Occasionally, the QUANT will expect you know whether a larger two-digit number, like

To see whether a number less than

How does one calculate the prime factorization of a number? In grade school, you may remember making "**factor trees**": that's the idea. To find the prime factorization of, for example,

Typically, once we are done, we sort the prime factors in numerical order:

Once we have the prime factorization, what can we do with it? See the next two items.

Suppose the QUANT asks: how many factors does

Each prime factor has an exponent (the exponent of

To find the total number of factors:

a) Find the list of exponents in the prime factorization — here

b) Add one to each number on the list — here

c) Multiply those together —

The number

Suppose the QUANT asked the number of odd factors of

This also means it has

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(Note:

Suppose a QUANT Math question involves finding, say, the

Find the prime factorizations of the two numbers:

Find the factors they have in common – the product of these is the

Express each number as the

GCFO1 = other stuff from first number

GCFO2 = other stuff from the second number

The

Let's do one more, just for practice. Suppose, on a QUANT math problem, we need to find the

Step (a):

Step (b):

Step (c)

Step (d)

1) The number of boxes in a warehouse can be divided evenly into

(A)

(B)

(C)

(D)

(E)

2) How many odd factors does

(A)

(B)

(C)

(D)

(E)

3) If n is the smallest integer such that

(A)

(B)

(C)

(D)

(E)

4) How many distinct prime numbers are factors of

(A) Four

(B) Five

(C) Six

(D) Seven

(E) Eight

5) If n is a positive integer, then(A) even only when

(B) odd only when

(C) odd only when

(D) always divisible by

(E) always one less than a prime number

1) C

2) E

3) B

4) B

5) D

1) This tells us that the number of boxes is evenly divisible by both

Step (a):

Step (b):

Step (c):

Step (d)

Thus,

**Answer: C**

2) Start with the prime factorization:

For odd factors, we put aside the factor of two, and look at the other prime factors.

Therefore, there are

**Answer: E**

3) The prime factorization of a square has to have even powers of all its prime factors. If the original number has a factor, say of

The factor of

**Answer: B**

4) Start with the prime factorization:

There are five distinct prime factors, {

**Answer: B**

5) Notice that

If n is even, then this product will be

If n is odd, this this product will be

No matter what, the product is even. Therefore, answers (A) & (B) & (C) are all out.

Let's look at a couple examples, to get a feel for this

Notice that one of the three numbers always has to be a multiple of

Therefore, Answer: D.

BTW, for answer choice E of that question, you will notice that for some trios of positives integers, adding one to the product does result in a prime, but for others, it doesn't.

This is a mathematical idea far far more advanced than anything on the QUANT, but it is mathematically impossible to create an easy rule or formula that will always result in prime numbers. The prime numbers follow an astonishingly complicated pattern, which is the subject of the single hardest unanswered question in modern mathematics: the Riemann Hypothesis.