Elite Mathematics / Linear / Topics / Prime Factors, HCF and LCM IGCSE · 4MA1 · Topic Prime Factors, HCF and LCM: past paper questions & worked solutions
Every Prime Factors, HCF and LCM question from the 4MA1 papers in our bank, 2020–2026, sorted from easiest to hardest. Try each one first, then open the worked solution.
26 questions
79 marks in total
48% of papers (26 of 54) include it
7/19/0 easy / medium / hard
Questions, easiest first Level is estimated from marks: 1–2 easy, 3–4 medium, 5+ hard.
1. Jun 2026 P2H · Q11 Easy 2 marksShow worked solution Solution | Step 1 \[3.6\times10^{24}=36\times10^{23}=(2^2\times3^2)(2\times5)^{23}=2^{25}3^2 5^{23}.\]
Rewrite with an integer coefficient, then decompose 36 and 10 into primes.
Solution | Step 2 The only prime factors are \(2,3,5\), so the highest is \(5\).
A larger composite factor is not a prime factor.
2. Jun 2025 P1HR · Q1 Easy 2 marksShow worked solution Find the LCM \[
45=3^2\times 5
\]
\[
70=2\times 5\times 7
\]
Find the LCM For the LCM, take the highest power of each prime:
\[
\text{LCM}=2\times 3^2\times 5\times 7=630
\]
3. Nov 2024 P1H · Q6 Easy 2 marksShow worked solution Find the LCM \[
A=2^3\times 5^4\times 7\times 11
\]
\[
B=2^2\times 5^2\times 7^2
\]
\[
C=2^2\times 5^3\times 7^4
\]
Find the LCM The HCF of \(A\), \(B\), and \(C\) uses the smallest power of each prime common to all three.
\[
2^2,\quad 5^2,\quad 7
\]
Find the LCM The prime \(11\) is not in all three numbers, so it is not included.
\[
\text{HCF}=2^2\times 5^2\times 7
\]
Final answer \(\displaystyle 2^2\times 5^2\times 7\)
4. May 2024 P1H · Q3 Easy 2 marksShow worked solution Find the LCM \[
72=2^3\times 3^2
\]
\[
108=2^2\times 3^3
\]
Find the LCM For the HCF, take the smaller power of each common prime:
\[
\text{HCF}=2^2\times 3^2
\]
\[
2^2\times 3^2=4\times 9=36
\]
5. Nov 2023 P1H · Q7 Easy 2 marksShow worked solution Find the LCM \[
A=5^3\times 7^3\times 11^6
\]
\[
B=5^6\times 7^2\times 11^4
\]
The HCF takes the smaller The HCF takes the smaller power of each common prime:
\[
\text{HCF}=5^3\times 7^2\times 11^4
\]
Final answer \(\displaystyle 5^3\times 7^2\times 11^4\)
6. Jan 2021 P2H · Q9 Easy 2 marksShow worked solution Find the LCM \[
A=2^3\times 3^2\times 5^2\times 11
\]
\[
B=2^4\times 3\times 5^4\times 13
\]
Find the LCM For the LCM, take the highest power of every prime that appears:
\[
\text{LCM}=2^4\times 3^2\times 5^4\times 11\times 13
\]
Final answer \(\displaystyle 2^4\times 3^2\times 5^4\times 11\times 13\)
7. Jan 2020 P1H · Q2 Easy 2 marksShow worked solution Write both numbers as products Write both numbers as products of prime factors:
\[
28=2^2\times 7
\]
\[
105=3\times 5\times 7
\]
Find the LCM For the lowest common multiple, take the highest power of each prime:
\[
\text{LCM}=2^2\times 3\times 5\times 7=420
\]
8. Jun 2025 P2HR · Q2 Medium 3 marksShow worked solution Find the LCM \[
1400=14\times 100
\]
\[
1400=(2\times 7)(2^2\times 5^2)
\]
\[
1400=2^3\times 5^2\times 7
\]
Final answer \(\displaystyle 2^3\times 5^2\times 7\)
9. May 2025 4WM2HR · Q2 Medium 3 marksShow worked solution Required Write 1400 as a product of powers of its prime factors.
Divide repeatedly by the smallest Divide repeatedly by the smallest prime factor each time: $$1400 \\;\\xrightarrow{\\div 2}\\; 700 \\;\\xrightarrow{\\div 2}\\; 350 \\;\\xrightarrow{\\div 2}\\; 175 \\;\\xrightarrow{\\div 5}\\; 35 \\;\\xrightarrow{\\div 5}\\; 7 \\;\\xrightarrow{\\div 7}\\; 1$$
Collect the prime factors Collect the prime factors: $$1400 = \\underbrace{2 \\times 2 \\times 2}\_{2^3} \\times \\underbrace{5 \\times 5}\_{5^2} \\times 7$$
Final answer 1400 = 2<sup>3</sup> × 5<sup>2</sup> × 7
10. Jun 2024 P2HR · Q1 Medium 3 marksShow worked solution Find the LCM \[
1400=14\times 100
\]
\[
1400=(2\times 7)(2^2\times 5^2)
\]
\[
1400=2^3\times 5^2\times 7
\]
Final answer \(\displaystyle 2^3\times 5^2\times 7\)
11. May 2023 P1HR · Q1 Medium 3 marksShow worked solution Find the LCM \[
2250=225\times 10
\]
\[
225=3^2\times 5^2,\qquad 10=2\times 5
\]
Find the LCM So
\[
2250=3^2\times 5^2\times 2\times 5
\]
\[
2250=2\times 3^2\times 5^3
\]
Final answer \(\displaystyle 2\times 3^2\times 5^3\)
12. Jun 2022 P2HR · Q2 Medium 3 marksShow worked solution Find the LCM \[
1200=12\times 100
\]
\[
1200=(2^2\times 3)(2^2\times 5^2)
\]
\[
1200=2^4\times 3\times 5^2
\]
Final answer \(\displaystyle 2^4\times 3\times 5^2\)
13. Jun 2022 P1HR · Q12 Medium 3 marksShow worked solution Find the LCM \[
P=3^3\times 5^2\times 7
\]
\[
Q=3^2\times 5\times 7^2
\]
Find the LCM For the HCF, take the smaller power of each common prime:
\[
\text{HCF}=3^2\times 5\times 7
\]
Find the LCM For \(P^3\times Q\):
\[
P^3=(3^3\times 5^2\times 7)^3=3^9\times 5^6\times 7^3
\]
Find the LCM Now multiply by \(Q\):
\[
P^3\times Q=(3^9\times 5^6\times 7^3)(3^2\times 5\times 7^2)
\]
\[
P^3\times Q=3^{11}\times 5^7\times 7^5
\]
Final answer (a) \(3^2\times 5\times 7\), (b) \(3^{11}\times 5^7\times 7^5\)
14. Jan 2022 P2H · Q7 Medium 3 marksShow worked solution Find the LCM \[
28=2^2\times 7
\]
\[
42=2\times 3\times 7
\]
\[
63=3^2\times 7
\]
Find the LCM For the LCM, take the highest power of each prime:
\[
\text{LCM}=2^2\times 3^2\times 7=252
\]
15. Nov 2021 P1H · Q7 Medium 3 marksShow worked solution Find the LCM \[
3.6\times 10^3=3600
\]
\[
3600=36\times 100
\]
\[
36=2^2\times 3^2,\qquad 100=2^2\times 5^2
\]
Find the LCM So
\[
3600=2^4\times 3^2\times 5^2
\]
Final answer \(\displaystyle 2^4\times 3^2\times 5^2\)
16. May 2021 P2H · Q1 Medium 3 marksShow worked solution Find the LCM \[
600=6\times 100
\]
\[
600=(2\times 3)(2^2\times 5^2)
\]
\[
600=2^3\times 3\times 5^2
\]
Final answer \(\displaystyle 2^3\times 3\times 5^2\)
17. Jan 2020 P2HR · Q1 Medium 3 marksShow worked solution Use the index law Use the index law \(a^m\times a^n=a^{m+n}\):
\[
5^{17}\times 5^2=5^{19}
\]
Find the LCM Now write \(800\) as a product of prime factors:
\[
800=8\times 100
\]
\[
800=2^3\times 2^2\times 5^2
\]
\[
800=2^5\times 5^2
\]
Final answer (a) \(5^{19}\), (b) \(2^5\times 5^2\)
18. May-Nov 2020 P1H · Q10 Medium 3 marksShow worked solution Find the LCM \[
A=2\times 3^{43}
\]
\[
B=16\times 3^{37}=2^4\times 3^{37}
\]
Find the LCM For the HCF, take the smaller powers:
\[
\text{HCF}=2\times 3^{37}
\]
Find the LCM For \(A\times B\),
\[
A\times B=(2\times 3^{43})(2^4\times 3^{37})
\]
\[
A\times B=2^5\times 3^{80}
\]
Final answer (a) \(2\times 3^{37}\), (b) \(2^5\times 3^{80}\)
19. May 2025 4WM2H · Q17 Medium 4 marksShow worked solution (a) \[
ABC=(3^2\times5^4)(2^5\times3^2)(2^3\times5^6)
\]
\[
=2^8\times3^4\times5^{10}
\]
\[
(ABC)^3=2^{24}\times3^{12}\times5^{30}
\]
(b) \[
(5\times10^{150})(3\times10^{140})=15\times10^{290}
\]
\[
=3\times5\times(2\times5)^{290}
\]
\[
=2^{290}\times3\times5^{291}
\]
Final answer \(2^{24}\times3^{12}\times5^{30}\); \(2^{290}\times3\times5^{291}\).
20. May 2024 P1HR · Q11 Medium 4 marksShow worked solution Find the LCM \[
A=2^5\times 5\times 7^2
\]
\[
B=2^3\times 5^3\times 7^4
\]
Find the LCM For part (a),
\[
5A=2^5\times 5^2\times 7^2
\]
\[
2B=2^4\times 5^3\times 7^4
\]
The HCF takes the smaller The HCF takes the smaller powers:
\[
\text{HCF}=2^4\times 5^2\times 7^2
\]
Find the LCM For part (b),
\[
AB=2^{5+3}\times 5^{1+3}\times 7^{2+4}
\]
\[
AB=2^8\times 5^4\times 7^6
\]
Find the LCM Therefore
\[
(AB)^2=2^{16}\times 5^8\times 7^{12}
\]
Final answer (a) \(2^4\times 5^2\times 7^2\), (b) \(2^{16}\times 5^8\times 7^{12}\)
21. Jun 2023 P2HR · Q2 Medium 4 marksShow worked solution Find the LCM Pens come in packs of 9 and pencils come in boxes of 12.
Find the LCM The smallest equal number of pens and pencils is
\[
\operatorname{LCM}(9,12)=36
\]
Find the LCM So Sandeep buys
\[
4\text{ packs of pens}
\]
Find the LCM and
\[
3\text{ boxes of pencils}
\]
Cost Cost:
\[
4(7.60)+3(4.80)=30.40+14.40=44.80
\]
Final answer \(\pounds44.80\).
22. May 2023 P1H · Q8 Medium 4 marksShow worked solution Find the LCM First write \(300\) as a product of prime factors:
\[
300=3\times 100=3\times 2^2\times 5^2
\]
\[
300=2^2\times 3\times 5^2
\]
Find the LCM Now
\[
A=2^3\times 3^2\times 5
\]
Find the LCM and
\[
B=2^2\times 3^3\times 5
\]
Find the LCM So
\[
5A=2^3\times 3^2\times 5^2
\]
Find the LCM and
\[
7B=2^2\times 3^3\times 5\times 7
\]
Find the LCM For the LCM of \(5A\) and \(7B\), take the highest power of every prime:
\[
\text{LCM}=2^3\times 3^3\times 5^2\times 7
\]
Final answer (a) \(2^2\times 3\times 5^2\), (b) \(2^3\times 3^3\times 5^2\times 7\)
23. Jun 2022 P1H · Q3 Medium 4 marksShow worked solution Find the LCM For the HCF of \(56\) and \(84\):
\[
56=2^3\times 7
\]
\[
84=2^2\times 3\times 7
\]
Take the smaller powers Take the smaller powers:
\[
\text{HCF}=2^2\times 7=28
\]
Find the LCM For the LCM of \(60\) and \(72\):
\[
60=2^2\times 3\times 5
\]
\[
72=2^3\times 3^2
\]
Take the larger powers Take the larger powers:
\[
\text{LCM}=2^3\times 3^2\times 5=360
\]
Final answer (a) \(28\), (b) \(360\)
24. Jan 2022 P1HR · Q6 Medium 4 marksShow worked solution Find the LCM \[
36=2^2\times 3^2
\]
\[
120=2^3\times 3\times 5
\]
Find the LCM For the LCM, take the highest power of each prime:
\[
\text{LCM}=2^3\times 3^2\times 5=360
\]
Find the LCM For part (b),
\[
A=5^2\times 7^4\times 11^p
\]
\[
B=5^m\times 7^{n-5}\times 11
\]
Find the LCM Since \(m>2\), the smaller power of \(5\) is \(5^2\).
Find the LCM Since \(n>10\), \(n-5>5\), so the smaller power of \(7\) is \(7^4\).
Find the LCM Since \(p>1\), the smaller power of \(11\) is \(11^1\).
\[
\text{HCF}=5^2\times 7^4\times 11
\]
Final answer (a) \(360\), (b) \(5^2\times 7^4\times 11\)
25. Jan 2021 P1HR · Q9 Medium 4 marksShow worked solution Find the LCM \[
A=2^8\times 3^5\times 11^4
\]
\[
B=2^6\times 3\times 11^8
\]
Find the LCM For the HCF, take the smaller power of each common prime:
\[
\text{HCF}=2^6\times 3\times 11^4
\]
Find the LCM Now
\[
2A=2^9\times 3^5\times 11^4
\]
Find the LCM and
\[
3B=2^6\times 3^2\times 11^8
\]
Find the LCM For the LCM of \(2A\) and \(3B\), take the larger power of each prime:
\[
\text{LCM}=2^9\times 3^5\times 11^8
\]
Final answer (a) \(2^6\times 3\times 11^4\), (b) \(2^9\times 3^5\times 11^8\)
26. Specimen 4WM2H · Q2 Medium 4 marksShow worked solution Find the LCM \[
56=2^3\times 7,\qquad 84=2^2\times 3\times 7
\]
The common prime factors are The common prime factors are \(2^2\) and \(7\), so
\[
\text{HCF}=2^2\times 7=28
\]
Find the LCM Also
\[
60=2^2\times 3\times 5,\qquad 72=2^3\times 3^2
\]
Find the LCM Use the highest powers of each prime:
\[
\text{LCM}=2^3\times 3^2\times 5=360
\]
Final answer HCF \(=28\), LCM \(=360\).
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