Factors & Multiples - 7 Types Of Commonly Tested Exam Questions
In this lesson, we will cover 7 commonly tested types of Factors and Multiples questions. Each question type is frequently examined, and by mastering all seven, students will be well-prepared to recognise and tackle similar questions confidently in exams, greatly increasing their chances of scoring well.
Related Lessons:
Type 1: Product Of Prime Factors
Express 9604 as a product of its prime factors.
Solution:
We can break 9604 down using the tree structure method or long division method, and we will get the answer:
\(9604=2^2\times7^4\)
Type 2: Square/Cube Root
Given that \(9604=2^2×7^4\), evaluate \(\sqrt{960400}\) using prime factorisation, without using a calculator.
Solution:
Now the approach is to first express 9604 as a product of its prime factors.
\[
\begin{align*}
\sqrt{960400}&=\sqrt{9604\times100}\\
&=\sqrt{2^2\times7^4}\times\sqrt{{10}^2}\\
&=2\times7^2\times10\\
&=980
\end{align*}
\]
Type 3: Find HCF & LCM
The numbers 42 and 90, expressed as product of their prime factors, are given as \(42=2×3×7\) and \(90=2×3^2×5\). Find the HCF & LCM of 42 and 90.
Solution:
To find the HCF, we select the common factors and select the smallest. Therefore,
\(HCF=2\times3=6\)
To find the LCM, we select the largest of all the prime factors. Therefore,
\(LCM=2\times3^2\times5\times7=630\)
Type 4: Perfect Square/Cube
The numbers 90, expressed as product of its prime factors, is \(90=2×3^2×5\). Find the smallest integer k, such as \(90k\) is a perfect square.
Solution:
\[
\begin{align*}
90&=2\times3^2\times5\\
90k&=2^2\times3^2\times5^2\\
k&=2\times5=10\\
\end{align*}
\]
Type 5: One Number Multiple Of Another
The numbers 42 and 90, expressed as product of their prime factors, are given as \(42=2×3×7\) and \(90=2×3^2×5\). Find the smallest positive integer value of p for which \(42p\) is a multiple of 90.
Solution:
To find the value of p, after dividing 42p with 90, the value of p is the value of the denominator after performing cancellation.
\[
\begin{align*}
\frac{42p}{90}&=\frac{2\times3\times7\times{p}}{2\times3^2\times5}\\
p&=3\times5=15\\
\end{align*}
\]
Type 6: HCF/LCM Word Problem Sum
HCF Problem Sum:
Find the largest possible number of groups if 150 boys and 120 girls are to be divided equally.
Solution:
Now, since 150 boys and 120 girls are divided into smaller groups, we are expecting a smaller number, so we will find the LCM. First, we will break down 150 and 120 into product of their prime factors.
\(120=2^3\times3\times5\)
\(150=2\times3\times5^2\)
To find the HCF, we select common and select the smallest prime factors. Therefore,
\(HCF=2\times3\times5=30\)
Largest group \(= \frac{150+120}{30}=9\) groups
LCM Problem Sum:
3 drivers can complete one round of the track in 60 seconds, 120 seconds and 140 seconds. If they start together, when will all three drivers meet again?
Solution:
We are looking at a future timing, which is a larger number. So we will find the LCM. First, we break down 60, 120 and 140 into product of their prime factors.
\(60=2^2\times3\times5\)
\(120=2^3\times3\times5\)
\(140=2^2\times5\times7\)
To find the LCM, we select all prime factors and select the largest. Therefore, the LCM is:
LCM\(=2^3\times3\times5\times7=840\) seconds
Type 7: Finding The Unknown Composite Number Given The HCF & LCM
Product Of Two Numbers = HCF × LCM
\[ \begin{align*} 126\times P&=18\times1260\\ P&=\frac{18\times1260}{126}\\ &=180\\ \end{align*} \]Watch Full Concept Breakdown With Examples
In this video, we will go through the complete solutions to 7 commonly tested question types on Factors & Multiples that we discuss above, with clear, step-by-step working shown for every example.