Factors & Multiples Practice 2
Tackle 6 carefully selected questions that focus on finding the HCF and LCM using the selection method. These examples reflect the types of problems most commonly seen in exams. Master these questions, and you’ll be well-prepared to handle similar problems with confidence on exam day.
Related Lessons:
Practice 2 - Question 1
Given that \(1960=2^3\times5\times7^2\) and \(2100=2^2\times3\times5^2\times7\), find the LCM of 1960 and 2100.
Solution:
\[
\begin{align*}
1960&=2^3\times3\times7^2\\
2100&=2^2\times3\times5^2\times7\\
LCM&=2^3\times3\times5^2\times7^2\\
&=29400\\
\end{align*}
\]
Practice 2 - Question 2
Given that \(1350=2\times3^3\times5^2\), and that \(\frac{1350}{k}\) is a square number, write down the smallest possible integer value of k.
Solution:
\[
\begin{align*}
1350&=2\times3^3\times5^2\\
\frac{1350}{k}&=\frac{2\times3^3\times5^2}{k}\\
&=\frac{2\times3\times3^2\times5^2}{k}\\
\end{align*}
\]
Now, \(\frac{1350}{k}\) is a perfect square. Since \(3^2\times5^2\) is a perfect square, therefore \(k=2\times3=6\).
Practice 2 - Question 3
Given that A and B are written as a product of their prime factors.
\[
\begin{align*}
A&=2^3\times3^2\times5\\
B&=2^2\times5^2\times11\\
\end{align*}
\]
Find the smallest positive integer n for which nB is a multiple of A.
Solution:
\[
\begin{align*}
\frac{nB}{A}&=\frac{2^2\times5^2\times11\times n}{2^3\times3^2\times5}\\
&=\frac{5\times11\times n}{2\times3^2}\\
\end{align*}
\]
Therefore, \(n={2\times3}^2=18\)
Practice 2 - Question 4
Ben and Jack jog on a circular track with a radius of 15 metres. Ben jogs with a constant speed of \(0.15\pi\ m/s\) and Jack jogs with a constant speed of \(0.25\pi\ m/s\). If both boys start jogging in the opposite direction from point A at 0810hours, when will they meet again at A?
Solution:
\[
\begin{align*}
Circumference&=2\pi r\\
&=2\pi\times15\\
&=30\pi\ metres\\
\end{align*}
\]
Since there are going to meet in the future, we are going to find the LCM.
\[
\begin{align*}
200&=2^3\times5^2\\
120&=2^2\times3\times5\\
LCM&=2^3\times3\times5^2\\
&=600 \ seconds\\
&=10 \ minutes
\end{align*}
\]
Ben and jack will meet 10 mins later at 0820hrs
Practice 2 - Question 5
Megan has 144 sweets, 120 chocolate bars and 42 lollipops. She packed the snacks into smaller bags with the same number of each type of snack. Find the greatest number of bags she can pack.
Solution:
Since we are going to find a smaller number, we will find the HCF.
\[
\begin{align*}
144&=2^4\times3^2\\
120&=2^3\times3\times5\\
42&=2\times3\times7\\
HCF&=2\times3\\
&=6
\end{align*}
\]
Practice 2 - Question 6
Given that \(m=2^2\times3^3\times5^k\) and \(\ n=2^3\times3^2\times5\), find the value of k if the LCM of m and n is 5400.
Solution:
\[
\begin{align*}
m&=2^2\times3^3\times5^k\\
n&=2^2\times3^3\times5\\
LCM&=2^2\times3^3\times5^k\\
5400&=2^2\times3^3\times5^2\\
\end{align*}
\]
By comparison, k=2.
Watch Full Solution Step-By-Step
To gain a clearer understanding of how to solve the questions above, it is helpful to follow the solutions step by step. The video below provides a detailed walkthrough, showing clearly how each solution is developed.