Are you confident enough to save these basic questions?

**“How many odd factors does 210 have?”**

** “If n is the smallest integer such that 432 times n is the square of an integer, what is the value of y?”**

** “How many prime numbers are factors of 33150?”**

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.

**Idea #1: Prime Numbers**

This is probably review, but just for a refresher: a prime number is any positive integer that is divisible by only 1 and itself. In other words, a prime number has only two factors: itself and 1. Numbers that have more than two factors are called composite. (By mathematical convention, 1 is the only positive integer considered neither prime nor composite.) The first few prime numbers are:

**2 3 5 7 11 13 17 19 23 29**

In preparation for the exams, it would be good to be familiar with this list. If you verify for yourself why each number from 2 to 30 is prime or composite, it will help you remember this list.

Occasionally, the exam will expect you know whether a larger two-digit number, like 67, is prime. Of course, if the number is even, it’s not prime. If it ends in a digit of 5, it’s divisible by five. For divisibility by 3, a good trick to know: if the sum of the digits is divisible by three, then the number is divisible by three. Here 6 + 7 = 13, not divisible by three, so 67 is not divisible by three.

To see whether a number less than 100 is prime, all we have to do is see whether it is divisible by one of the single digit prime numbers: 2, 3, 5, or 7. We’ve already checked 2, 3, and 5. The number 67 is not divisible by 7: 7 goes evenly into 63 and 70, not 67. That’s enough checking to establish irrevocably that 67 is prime.

**Idea #2: Prime Factorization**

Every positive integer greater than 1 can be written in a unique way as a product of prime numbers; this is called its prime factorization. The prime factorization is analogous to the DNA of the number, the unique blueprint by which to construct the number. In other words, when you calculate the prime factorization of a number, you have some powerful information at your disposal.

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, 48, we simply choose any two factors — say 6 and 8 — and then choose factors of those number, and then of those numbers, until we are left with nothing but primes.

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

48 = 2*2*2*2*3 = (2^4)(3)

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

**Idea #3: The Number of Factors**

Suppose the exam asks: how many factors does 1440 have? It would be quite tedious to count them all, but there’s a fast trick once you have the prime factorization. First of all, the prime factorization of 1440 is

1440 = (2^5)*(3^2)*5

Each prime factor has an exponent (the exponent of 5 is 1).

To find the total number of factors:

a) Find the list of exponents in the prime factorization — here {5, 2, 1}

b) Add one to each number on the list — here {6, 3, 2}

c) Multiply those together — 6*3*2 = 36

The number 1440 has thirty-six factors, including 1 and itself.

Suppose the exam asked the number of odd factors of 1440. We know that odd factors cannot contain any factor of 2 at all, so basically we repeat that procedure with all the factors except the factors of 2. Here: {2, 1} –> {3, 2} –> 3*2 = 6. The number 1440 has 6 odd factors, including 1. Just for verification, the odd factors of 1440 are

{1, 3, 5, 9, 15, and 45}

This also means it has 36 – 6 = 30 even factors.

**Idea #4: HCF and LCM**

HCF= highest common factor

LCM = least common multiple.

(Note: LCM and LCD are the same thing: a least common denominator, LCD, of two number is always their LCM.)

Suppose a Math question involves finding, say, the LCM (or LCD) of 30 and 48. There’s a very straightforward procedure to find the LCM.

Find the prime factorizations of the two numbers: 30 = 2*3*5 and 48 = 2*2*2*2*3

Find the factors they have in common – the product of these is the GCF. Here, the HCF = 2*3 = 6

Express each number as the HCF*(other stuff): 30 = 6*5 and 48 = 6*8

The LCM = HCF*(other stuff from first number)*(other stuff from the second number): LCM = 6*5*8 = 240

Let’s do one more, just for practice. Suppose, on a math problem, we need to find the LCM/LCD of 28 and 180

Step (a): 28 = 2*2*7, 180 = 2*2*3*3*5

Step (b): 28 = 2*2*7, 180 = 2*2*3*3*5; GCF = 2*2 = 4

Step (c) 28 = 4*7, 180 = 4*45

Step (d) LCM = 4*7*45 = 1260

Check out the next article for more practice question and shortcuts to solve faster and improve your accuracy .

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