Lowest common multiple questions with answers

Questions

1. Question 1LCM of two numbers

   Find the lowest common multiple (LCM) of $72$ and $96$.

   Show the worked solution

   1. Step 1: Short division of 72 by primes, smallest first, until you reach 1; then 72 in index notation$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;72\;} \\ 2 & \underline{\;36\;} \\ 2 & \underline{\;18\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$$72 = 2^{3} \times 3^{2}$
   2. Step 2: Short division of 96 by primes, smallest first, until you reach 1; then 96 in index notation$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;96\;} \\ 2 & \underline{\;48\;} \\ 2 & \underline{\;24\;} \\ 2 & \underline{\;12\;} \\ 2 & \underline{\;6\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$$96 = 2^{5} \times 3$
   3. Step 3: LCM: decide which primes to take, and which power of eachTake every prime that appears, each with its highest power.
   4. Step 4: Write the LCM in index notation, then work it outLCM $= 2^{5} \times 3^{2}$$= 288$
2. Question 2LCM of three numbers

   Find the lowest common multiple (LCM) of $18$, $30$ and $42$.

   Show the worked solution

   1. Step 1: Short division of 18 by primes, smallest first, until you reach 1$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;18\;} \\ 3 & \underline{\;9\;} \\ 3 & \underline{\;3\;} \\ & \;1 \end{array}$
   2. Step 2: Short division of 30 by primes, smallest first, until you reach 1$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;30\;} \\ 3 & \underline{\;15\;} \\ 5 & \underline{\;5\;} \\ & \;1 \end{array}$
   3. Step 3: Short division of 42 by primes, smallest first, until you reach 1$\def\arraystretch{1.8}\begin{array}{r|l} 2 & \underline{\;42\;} \\ 3 & \underline{\;21\;} \\ 7 & \underline{\;7\;} \\ & \;1 \end{array}$
   4. Step 4: Count how many times each prime divides: the numbers in index notation$18 = 2 \times 3^{2}$$30 = 2 \times 3 \times 5$$42 = 2 \times 3 \times 7$
   5. Step 5: LCM: decide which primes to take, and which power of eachTake every prime that appears, each with its highest power.
   6. Step 6: Write the LCM in index notation, then work it outLCM $= 2 \times 3^{2} \times 5 \times 7$$= 630$
3. Question 3LCM in index notation

   It is given that $360 = 2^{3} \times 3^{2} \times 5$ and $756 = 2^{2} \times 3^{3} \times 7$. Find the lowest common multiple (LCM) of $360$ and $756$, giving your answer in index notation.

   Show the worked solution

   1. Step 1: LCM: decide which primes to take, and which power of eachTake every prime that appears, each with its highest power.
4. Question 4Smallest n so that nA is a multiple of B

   It is given that $100 = 2^{2} \times 5^{2}$ and $315 = 3^{2} \times 5 \times 7$. Find the smallest positive whole number $n$ such that $100n$ is a multiple of $315$.

   Show the worked solution

   1. Step 1: 100n is a multiple of 315 when it has every prime of 315, each to at least the same power: compare the powers prime by prime$3$: $315$ has $3^{2}$, $100$ has none$5$: $315$ has $5^{1}$, $100$ has $5^{2}$: enough$7$: $315$ has $7^{1}$, $100$ has none
   2. Step 2: n supplies the missing powers only$n = 3^{2} \times 7$$= 63$Check: $100n = 2^{2} \times 3^{2} \times 5^{2} \times 7$, a multiple of $315$.
5. Question 5Unknown powers from the HCF and LCM

   Two numbers, $A$ and $B$, are written as products of their prime factors: $A = 2 \times 3^{2} \times 7^{a}$ and $B = 2^{b} \times 7^{2}$, where $a$ and $b$ are positive whole numbers. The HCF of $A$ and $B$ is $2 \times 7$ and their LCM is $2^{3} \times 3^{2} \times 7^{2}$. Find the value of $a$ and the value of $b$.

   Show the worked solution

   1. Step 1: HCF: the LOWER power of each common prime. LCM: the HIGHER power of every prime. Compare the powers of each prime with an unknownPrime $7$: the HCF has $7^{1}$, the LOWER power; B has $7^{2}$, so the lower one is A's.$a = 1$Prime $2$: the LCM has $2^{3}$, the HIGHER power; A has $2^{1}$, so the higher one is B's.$b = 3$