Sec 1 Maths · Primes, HCF and LCM

Prime numbers and prime factorisation questions with answers

Practice questions on prime numbers and prime factorisation, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

State whether $139$ is a prime number.

Step 1 of 3

Only primes up to the square root need testing: find the square root (to 2 decimal places), then list those primes

\sqrt{139} = 11.78\ldots

Primes to test:

2, 3, 5, 7, 11

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Questions

Question 1Is it a prime number?
State whether $71$ is a prime number.
Show the worked solution
1. Step 1: Only primes up to the square root need testing: find the square root (to 2 decimal places), then list those primes $\sqrt{71} = 8.42\ldots$ Primes to test: $2, 3, 5, 7$
2. Step 2: Try dividing by each of these primesNone of $2, 3, 5, 7$ divides $71$ exactly.
3. Step 3: A prime number has exactly two factors: 1 and itselfSo $71$ is a prime number.
Question 2Prime factorisation: two primes
Express $112$ as a product of its prime factors, giving your answer in index notation.
Show the worked solution
1. Step 1: Short division: divide by the smallest prime that goes in exactly, again and again, until you reach 1 $\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;112\;} \\ 2 & \underline{\;56\;} \\ 2 & \underline{\;28\;} \\ 2 & \underline{\;14\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}$
2. Step 2: Count how many times each prime divides, and write the product in index notation $112 = 2^{4} \times 7$
Question 3Prime factorisation: three or more primes
Express $1260$ as a product of its prime factors, giving your answer in index notation.
Show the worked solution
1. Step 1: Short division: divide by the smallest prime that goes in exactly, again and again, until you reach 1 $\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;1260\;} \\ 2 & \underline{\;630\;} \\ 3 & \underline{\;315\;} \\ 3 & \underline{\;105\;} \\ 5 & \underline{\;35\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}$
2. Step 2: Count how many times each prime divides, and write the product in index notation $1260 = 2^{2} \times 3^{2} \times 5 \times 7$
Question 4Is the statement right? Give the reason
Siti says, " $111$ is a prime number." Is Siti correct? Choose the answer with its reason.
Show the worked solution
1. Step 1: A prime number has exactly two different factors: 1 and itselfA prime number has exactly $2$ factors: $1$ and itself.
2. Step 2: Try dividing by the primes 2, 3, 5, 7, 11, 13 in turn $111 \div 3 = 37$
3. Step 3: Decide $111$ has the factors $1$ , $3$ , $37$ and $111$ , so it is not prime: Siti is not correct.

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Questions

Question 1Is it a prime number?

State whether $71$ is a prime number.

Show the worked solution

Step 1: Only primes up to the square root need testing: find the square root (to 2 decimal places), then list those primes $\sqrt{71} = 8.42\ldots$ Primes to test: $2, 3, 5, 7$
Step 2: Try dividing by each of these primesNone of $2, 3, 5, 7$ divides $71$ exactly.
Step 3: A prime number has exactly two factors: 1 and itselfSo $71$ is a prime number.

Question 2Prime factorisation: two primes

Express $112$ as a product of its prime factors, giving your answer in index notation.

Show the worked solution

Step 1: Short division: divide by the smallest prime that goes in exactly, again and again, until you reach 1 $\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;112\;} \\ 2 & \underline{\;56\;} \\ 2 & \underline{\;28\;} \\ 2 & \underline{\;14\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}$
Step 2: Count how many times each prime divides, and write the product in index notation $112 = 2^{4} \times 7$

Question 3Prime factorisation: three or more primes

Express $1260$ as a product of its prime factors, giving your answer in index notation.

Show the worked solution

Step 1: Short division: divide by the smallest prime that goes in exactly, again and again, until you reach 1 $\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;1260\;} \\ 2 & \underline{\;630\;} \\ 3 & \underline{\;315\;} \\ 3 & \underline{\;105\;} \\ 5 & \underline{\;35\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}$
Step 2: Count how many times each prime divides, and write the product in index notation $1260 = 2^{2} \times 3^{2} \times 5 \times 7$

Question 4Is the statement right? Give the reason

Siti says, " $111$ is a prime number." Is Siti correct? Choose the answer with its reason.

Show the worked solution

Step 1: A prime number has exactly two different factors: 1 and itselfA prime number has exactly $2$ factors: $1$ and itself.
Step 2: Try dividing by the primes 2, 3, 5, 7, 11, 13 in turn $111 \div 3 = 37$
Step 3: Decide $111$ has the factors $1$ , $3$ , $37$ and $111$ , so it is not prime: Siti is not correct.