From Zero to Engineer

Divisors, prime factorization, GCD and LCM

Review exercises on the order of operations and rounding, what divisors and prime numbers are, how to factorize a number into primes, and how to read the GCD and LCM from a factorization.

A divisor is a number that divides another without a remainder. Divisors lead to prime numbers, primes lead to factorization, and from a factorization you can read off the GCD and LCM. This lesson walks through that whole chain on concrete numbers.

Review exercises

1) Place the appropriate symbols (< or >) between each of the following pairs of numbers

-1 > -6

5 > -29

-14 < 7

2) Calculate the value of each of the following expressions

16 - 12 \cdot 4 + 8 : 2 = 16 - 48 + 4 = -32 + 4 = -28

(16 - 12) \cdot (4 + 8) : 2 = 4 \cdot 12 : 2 = 48 : 2 = 24

\begin{aligned}9 - 3[17 + 5(5 - 7)] &= 9 - 3[17 + 5 \cdot (-2)] \\&= 9 - 3[17 - 10] = 9 - 3 \cdot 7 \\&= 9 - 21 = -12\end{aligned}

\begin{aligned}8[3(2 + 4) - 2(5 + 7)] &= 8[3 \cdot 6 - 2 \cdot 12] \\&= 8[18 - 24] = 8 \cdot (-6) = -48\end{aligned}

3) Prove that

6 - (3 - 2) \neq (6 - 3) - 2

6 - 1 \neq 3 - 2 \quad\Rightarrow\quad 5 \neq 1

24 : (2 + 6) \neq (24 : 2) + (24 : 6)

24 : 8 \neq 12 + 4 \quad\Rightarrow\quad 3 \neq 16

100 : (10 : 5) \neq (100 : 10) : 5

100 : 2 \neq 10 : 5 \quad\Rightarrow\quad 50 \neq 2

24 : (2 - 6) \neq (24 : 2) - (24 : 6)

24 : (-4) \neq 12 - 4 \quad\Rightarrow\quad -6 \neq 8

4) Round each number to the nearest 10, 100, and 1,000

number	to 10	to 100	to 1,000
2562	2560	2600	3000
1500	1500	1500	2000
−3451	−3450	−3500	−3000
−14525	−14530	−14500	−15000

Factors and divisors

Factors are a pair of numbers whose product equals a given number.

3 \cdot 6 = 18

A divisor is an integer that divides another number without a remainder.

the divisors of $18$ are $1,\ 2,\ 3,\ 6,\ 9,\ 18$

the divisors of $12$ are $1,\ 2,\ 3,\ 4,\ 6,\ 12$

the divisors of $25$ are $1,\ 5,\ 25$

the divisors of $17$ are $1,\ 17$

Seventeen is an example of a prime number.

A prime number has exactly two divisors: 1 and itself. Other prime numbers are 2, 3, 5, 7, 11, 13 …

Prime factorization

Every natural number can be expressed as a product of prime factors only.

126	2
63	3
21	3
7	7
1

126 = 2 \cdot 3 \cdot 3 \cdot 7

84	2
42	2
21	3
7	7
1

84 = 2 \cdot 2 \cdot 3 \cdot 7

512	2
256	2
128	2
64	2
32	2
16	2
8	2
4	2
2	2
1

512 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

Greatest common divisor (GCD) and least common multiple (LCM)

144	2
72	2
36	2
18	2
9	3
3	3
1

144 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

66	2
33	3
11	11
1

66 = 2 \cdot 3 \cdot 11

\mathrm{GCD}(144;\ 66) = 2 \cdot 3 = 6

\mathrm{LCM}(144;\ 66) = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 11 = 1584

\mathrm{GCD}(84;\ 512) = 2 \cdot 2 = 4

\mathrm{LCM}(84;\ 512) = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 7 = 10752

Frequently asked questions

How do you factorize a number into primes?

Divide the number by successive prime numbers, starting from the smallest, writing each quotient underneath. Repeat until you reach one. The factors in the right column form the factorization, for example 126 = 2 · 3 · 3 · 7.

How do you find the GCD and LCM from a factorization?

The GCD is the product of the factors common to both numbers; the LCM is the product of all factors, taking each repeated factor as many times as it appears at most in one factorization. For 144 and 66 that gives a GCD of 6 and an LCM of 1584.

How many divisors does a prime number have?

Exactly two: one and itself. That is why 17 is prime and 12 is not, since it is also divisible by 2, 3, 4 and 6.

← Previous

Properties of operations: commutative, associative and distributive

Fractions: multiplying, dividing, adding and simplifying