Highest common factor questions with answers

Questions

1. Question 1HCF of two numbers

   Find the highest common factor (HCF) of $36$ and $45$.

   Show the worked solution

   1. Step 1: Short division of 36 by primes, smallest first, until you reach 1; then 36 in index notation$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;36\;} \\ 2 & \underline{\;18\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$$36 = 2^{2} \times 3^{2}$
   2. Step 2: Short division of 45 by primes, smallest first, until you reach 1; then 45 in index notation$\def\arraystretch{1.8}\begin{array}{r|l} 3 & \underline{\;45\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}$$45 = 3^{2} \times 5$
   3. Step 3: HCF: decide which primes to take, and which power of eachTake only the primes common to every number, each with its lowest power.
   4. Step 4: Write the HCF in index notation, then work it outHCF $= 3^{2}$$= 9$
2. Question 2HCF of two numbers

   Find the highest common factor (HCF) of $12$ and $20$.

   Show the worked solution

   1. Step 1: Short division of 12 by primes, smallest first, until you reach 1; then 12 in index notation$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$$12 = 2^{2} \times 3$
   2. Step 2: Short division of 20 by primes, smallest first, until you reach 1; then 20 in index notation$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;20\;} \\ 2 & \underline{\;10\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}$$20 = 2^{2} \times 5$
   3. Step 3: HCF: decide which primes to take, and which power of eachTake only the primes common to every number, each with its lowest power.
   4. Step 4: Write the HCF in index notation, then work it outHCF $= 2^{2}$$= 4$
3. Question 3HCF of three numbers

   Find the highest common factor (HCF) of $12$, $18$ and $30$.

   Show the worked solution

   1. Step 1: Short division of 12 by primes, smallest first, until you reach 1$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$
   2. Step 2: Short division of 18 by primes, smallest first, until you reach 1$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;18\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$
   3. Step 3: Short division of 30 by primes, smallest first, until you reach 1$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}$
   4. Step 4: Count how many times each prime divides: the numbers in index notation$12 = 2^{2} \times 3$$18 = 2 \times 3^{2}$$30 = 2 \times 3 \times 5$
   5. Step 5: HCF: decide which primes to take, and which power of eachTake only the primes common to every number, each with its lowest power.
   6. Step 6: Write the HCF in index notation, then work it outHCF $= 2 \times 3$$= 6$
4. Question 4HCF of three numbers

   Find the highest common factor (HCF) of $12$, $18$ and $27$.

   Show the worked solution

   1. Step 1: Short division of 12 by primes, smallest first, until you reach 1$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$
   2. Step 2: Short division of 18 by primes, smallest first, until you reach 1$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;18\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$
   3. Step 3: Short division of 27 by primes, smallest first, until you reach 1$\def\arraystretch{1.8}\begin{array}{r|l} 3 & \underline{\;27\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$
   4. Step 4: Count how many times each prime divides: the numbers in index notation$12 = 2^{2} \times 3$$18 = 2 \times 3^{2}$$27 = 3^{3}$
   5. Step 5: HCF: decide which primes to take, and which power of eachTake only the primes common to every number, each with its lowest power.
   6. Step 6: Write the HCF in index notation, then work it outHCF $= 3$$= 3$
5. Question 5HCF in index notation

   It is given that $756 = 2^{2} \times 3^{3} \times 7$ and $396 = 2^{2} \times 3^{2} \times 11$. Find the highest common factor (HCF) of $756$ and $396$, giving your answer in index notation.

   Show the worked solution

   1. Step 1: HCF: decide which primes to take, and which power of eachTake only the primes common to every number, each with its lowest power.
6. Question 6HCF in index notation

   It is given that $1400 = 2^{3} \times 5^{2} \times 7$ and $1176 = 2^{3} \times 3 \times 7^{2}$. Find the highest common factor (HCF) of $1400$ and $1176$, giving your answer in index notation.

   Show the worked solution

   1. Step 1: HCF: decide which primes to take, and which power of eachTake only the primes common to every number, each with its lowest power.