Question

Find the prime factors of the number 60.

Answer 1

Factors are the numbers that conjure the said number when multiplied together. However, the question asks for "Prime factors ". Prime factors are those which are factors of the quantity, but are all prime.

Complete step by step solution:
Coming to the ultimate basics of the number system, we know that prime numbers are positive integers that have exactly two different factors: itself and 1.
Every composite number, i.e., any number that is not prime or that has factors other than itself and 1, can be expressed as a product of prime factors. We can form a factor tree to find the prime actors of the number in the question.

$$\;\;\;\;\;60 \\\ \;\;\;\;/\;\;\;\backslash \\\ \;\;\;3\; \times \;20 \\\ \;\;\;/\;\;\;\;/\backslash \\\ \;\;3\; \times \;5 \times 4 \\\ \;/\;\;\;\;/\;\;\;/\backslash \\\ 3\; \times 5\; \times 2 \times 2 \\\ \;\;\;\;\;\; \\\$$

The last level shows all prime numbers that are factors of the number, 60.
The factor tree may also be formed as shown below.

$$\;\;\;\;\;60 \\\ \;\;\;\;/\;\;\;\backslash \\\ \;\;\;2\; \times \;30 \\\ \;\;\;/\;\;\;\;/\;\;\backslash \\\ \;\;2\; \times 15 \times 2 \\\ \;/\;\;\;\;/\;\backslash \;\;\;\;\backslash \\\ 2 \times \;5\; \times 3\; \times \;2 \\\ \;\;\;\;\;\; \\\$$

So, you can see, though the factor trees began differently, they ultimately are the product of the same prime factors with the order being different. Thus, the prime factors of 60 are 2, 3 and 5.

Alternate method: Instead of forming a factor tree, one might as well in this manner.
$60 = 2 \times 30 = 2 \times 15 \times 2 = {2^2} \times 5 \times 3$
Thus, we can see the prime factors of 60 are 2, 3 and 5.

Note: For larger numbers, the formation of the factor tree might be very lengthy based on the speed of a person’s calculating speed. You can use simple division in that case. Use the quotients thus and use divisibility rules to figure out where to start the division from.