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PSLE

PSLE Foundation Mathematics

James has 600 one-centimetre cubes. He arranges all the cubes into a cuboid whose base is a square. Find the smallest possible height of the cuboid.

Highest Common Factor and Its Applications

2025 NASS PRELIMS P1 Q5

Primes, HCF and LCM

<p>Trees will be planted along $2$ roads, Privet Road and Wisteria Road. Privet Road is $0.48$ km long, and Wisteria road is $792$ m long. There will be a tree planted at the beginning and at the end of each road. The distance between every $2$ consecutive trees on Privet Road is the same and this distance is also equal to the distance between every $2$ consecutive trees on Wisteria Road.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the maximum possible distance between $2$ consecutive trees on both roads such that the conditions above are satisfied.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the total number of trees that are needed for both roads. Assume that trees are planted only on one side of each road.</span></div></li></ol>

Trees will be planted along $2$ roads, Privet Road and Wisteria Road. Privet Road is $0.48$ km long, and Wisteria road is $792$ m long. There will be a tree planted at the beginning and at the end of each road. The distance between every $2$ consecutive trees on Privet Road is the same and this distance is also equal to the distance between every $2$ consecutive trees on Wisteria Road.

Find the maximum possible distance between $2$ consecutive trees on both roads such that the conditions above are satisfied.

Hence, find the total number of trees that are needed for both roads. Assume that trees are planted only on one side of each road.

IP1U1 SWIII Q9

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express $612$ as a product of its prime factors.</span></div></li><li><div class="list-item-content"><span class="list-item-text">$72$ adult volunteers, $612$ female students and $1152$ male students attended a camp. What is the greatest number of groups that can be formed if all adult volunteers, female students and male students have to be distributed equally among all the groups?</span></div></li><li><div class="list-item-content"><span class="list-item-text">How many adult volunteers will there be in each group?</span></div></li></ol>

Express $612$ as a product of its prime factors.

$72$ adult volunteers, $612$ female students and $1152$ male students attended a camp. What is the greatest number of groups that can be formed if all adult volunteers, female students and male students have to be distributed equally among all the groups?

How many adult volunteers will there be in each group?

Prime Factorisation of Positive Integers

IP1U1 SWIII Q8

<p>Mary has three ribbons of lengths $140$ cm, $224$ cm and $560$ cm respectively. She wishes to cut all the ribbons into maximum equal number of pieces without any leftover ribbons.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the total number of pieces of ribbons that Mary has cut.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Mary’s friend, Jane, asked for a piece of cut ribbon with $15$ cm length. Is Mary able to give Jane what she asked for? Explain.</div></li></ol>
<p>Each original ribbon is cut into the same number of equal pieces, with each piece a positive whole number of centimetres long. For part (b), do not join or cut any pieces further.</p>

Mary has three ribbons of lengths $140$ cm, $224$ cm and $560$ cm respectively. She wishes to cut all the ribbons into maximum equal number of pieces without any leftover ribbons.

Find the total number of pieces of ribbons that Mary has cut.

Mary’s friend, Jane, asked for a piece of cut ribbon with $15$ cm length. Is Mary able to give Jane what she asked for? Explain.

Each original ribbon is cut into the same number of equal pieces, with each piece a positive whole number of centimetres long. For part (b), do not join or cut any pieces further.

IP1U1 SWIII Q6

<p>John has $3$ pieces of ropes with lengths of $120$ cm, $180$ cm and $210$ cm. He wishes to cut the $3$ pieces of rope into smaller pieces of equal length without any remainder.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">What is the greatest possible length of each smaller piece of rope?</span></div></li><li><div class="list-item-content"><span class="list-item-text">How many smaller pieces of rope can he get altogether?</span></div></li></ol>

John has $3$ pieces of ropes with lengths of $120$ cm, $180$ cm and $210$ cm. He wishes to cut the $3$ pieces of rope into smaller pieces of equal length without any remainder.

What is the greatest possible length of each smaller piece of rope?

How many smaller pieces of rope can he get altogether?

IP1U1 SWIII Q4

<p>The school hall in Beatty Secondary School measures $198$ m by $156$ m. The school decides to lay the floor with complete square tiles. Find the</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">largest possible length of the side of the square tile.</span></div></li><li><div class="list-item-content"><span class="list-item-text">number of tiles needed.</span></div></li></ol>

The school hall in Beatty Secondary School measures $198$ m by $156$ m. The school decides to lay the floor with complete square tiles. Find the

largest possible length of the side of the square tile.

IP1U1 SWIII Q2

<p>Two integers, written as products of their prime factors, are $2^p \times 3 \times 5^r$ and $2 \times 3^q \times 5^4$.</p>
<p>The highest common factor of these two integers is $2 \times 3 \times 5^2$. The lowest common multiple of these two integers is $2^3 \times 3 \times 5^4$. Find the values of $p$, $q$ and $r$.</p>

Two integers, written as products of their prime factors, are $2^p \times 3 \times 5^r$ and $2 \times 3^q \times 5^4$.

The highest common factor of these two integers is $2 \times 3 \times 5^2$. The lowest common multiple of these two integers is $2^3 \times 3 \times 5^4$. Find the values of $p$, $q$ and $r$.

Lowest Common Multiple and Its Applications

IP1U1 SWII Q18

<p style="margin-bottom: 1.5rem">Written as the product of their prime factors,</p>
<p>First integer $ \qquad\qquad\quad 2^p \times 3^q$</p>
<p style="margin-bottom: 1.5rem">Second integer $ \qquad\qquad 2 \times 3 \times 5^r$</p>
<p>The highest common factor of these two integers is $2 \times 3$. The lowest common multiple of these two integers is $2^2 \times 3 \times 5$.</p>
<p>Find the values of $p$, $q$ and $r$.</p>

Written as the product of their prime factors,

First integer $ \qquad\qquad\quad 2^p \times 3^q$

Second integer $ \qquad\qquad 2 \times 3 \times 5^r$

The highest common factor of these two integers is $2 \times 3$. The lowest common multiple of these two integers is $2^2 \times 3 \times 5$.

Find the values of $p$, $q$ and $r$.

IP1U1 SWII Q17

<p>The prime factorisation of two numbers are $2^{5} \times 5^{10} \times 11 \times 13^{2}$ and $2^{3} \times 5 \times 11 \times 17^{3}$.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">HCF and,</span></div></li><li><div class="list-item-content"><span class="list-item-text">LCM of these two numbers, expressing your answers in index notation.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Find the smallest positive integer $k$ such that $\sqrt{2^{5} \times 5^{10} \times 11 \times 13^{2} \times k}$ is a whole number.</span></div></li></ol>

The prime factorisation of two numbers are $2^{5} \times 5^{10} \times 11 \times 13^{2}$ and $2^{3} \times 5 \times 11 \times 17^{3}$.

Find the smallest positive integer $k$ such that $\sqrt{2^{5} \times 5^{10} \times 11 \times 13^{2} \times k}$ is a whole number.

Perfect Squares and Cubes Using Prime Factors

IP1U1 SWII Q12

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the prime factorization of $2772$, expressing your answer in index notation.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Given that the prime factorization of $4200$ is $2^{3} \times 3 \times 5^{2} \times 7$, find the highest common factor of $2772$ and $4200$.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find the smallest positive integer $k$ such that&nbsp;$(2772 \times 4200k)$ is a square number.</div></li></ol>

Find the prime factorization of $2772$, expressing your answer in index notation.

Given that the prime factorization of $4200$ is $2^{3} \times 3 \times 5^{2} \times 7$, find the highest common factor of $2772$ and $4200$.

Find the smallest positive integer $k$ such that $(2772 \times 4200k)$ is a square number.

IP1U1 SWII Q11

<p>Expressed as the product of its prime factors, $3375=3^{3} \times 5^{3}$ and $32400=2^{4} \times 3^{4} \times 5^{2}$.</p>
<p>Using these results,</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">find in prime factorization form, the highest common factor of $3375$ and $32400$.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">find the value of $\sqrt{32400} \div \sqrt[3]{3375}$.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">find the smallest positive integer $k$, such that&nbsp;$3375k$ is a perfect square.</div></li></ol>

Expressed as the product of its prime factors, $3375=3^{3} \times 5^{3}$ and $32400=2^{4} \times 3^{4} \times 5^{2}$.

find in prime factorization form, the highest common factor of $3375$ and $32400$.

find the value of $\sqrt{32400} \div \sqrt[3]{3375}$.

find the smallest positive integer $k$, such that $3375k$ is a perfect square.

Square and Cube Roots Using Prime Factorisation

IP1U1 SWII Q10

<p>Given that $p=2^{3} \times 3^{4}$ and $q=2^{2} \times 3$ find the HCF of $p$ and $q$.</p>

Given that $p=2^{3} \times 3^{4}$ and $q=2^{2} \times 3$ find the HCF of $p$ and $q$.

IP1U1 SWII Q4

<p>Find the highest common factor of $72216$ and $324$.</p>

Find the highest common factor of $72216$ and $324$.

IP1U1 SWII Q3

<p>Find the HCF of $2^{3} \times 3 \times 5^{2}$ and $2^{3} \times 3^{2} \times 5 \times 7$.</p>

Find the HCF of $2^{3} \times 3 \times 5^{2}$ and $2^{3} \times 3^{2} \times 5 \times 7$.

IP1U1 SWII Q2

<p>Find the highest common factor of $180$ and $320$.</p>

Find the highest common factor of $180$ and $320$.

IP1U1 SWII Q1

<p>Find two integers, both smaller than $200$, that have a lowest common multiple of $600$ and a highest common factor of $20$.<span class="inline-marks">[2]</span></p>

Find two integers, both smaller than $200$, that have a lowest common multiple of $600$ and a highest common factor of $20$.

IP1 Class Test (May 2025) Q4

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express $3024$ as a product of its prime factors in index notation.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that $8640 = 2^6 \times 3^3 \times 5$, find the HCF and LCM of $3024$ and $8640$, giving your answer as the product of its prime factors in index notation.</span><span class="inline-marks">[2]</span></div></li></ol>

Express $3024$ as a product of its prime factors in index notation.

Given that $8640 = 2^6 \times 3^3 \times 5$, find the HCF and LCM of $3024$ and $8640$, giving your answer as the product of its prime factors in index notation.

IP1 Class Test (May 2025) Q3

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = n^2 + 1$.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Write down the first $5$ terms of the sequence.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the lowest common multiple of the $7^{th}$ and the $8^{th}$ term.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the highest common factor of the $4^{th}$ and the $13^{th}$ term.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">The first $5$ terms of another sequence are $4, 7, 12, 19, 28$.</span></div><ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">By comparing this sequence and the sequence in (i), find a formula for the $n^{th}$ term of this sequence in terms of $n$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the $21^{st}$ term of this sequence.</span></div></li></ol></li></ol>

Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = n^2 + 1$.

The first $5$ terms of another sequence are $4, 7, 12, 19, 28$.

General Term of a Sequence

Number Patterns

<p>Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = 7n + 5$, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the $2^{nd}$ term,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the sum of the $24^{th}$ and the $27^{th}$ term,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the highest common factor of the $10^{th}$ and the $15^{th}$ term.</span></div></li></ol>

Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = 7n + 5$, find

the sum of the $24^{th}$ and the $27^{th}$ term,

the highest common factor of the $10^{th}$ and the $15^{th}$ term.

 General Term of a Sequence

<p>Two numbers are such that their highest common factor is $150$ and lowest common multiple is $600$. Find all the possible two numbers if all the numbers are at least $120$.</p>

Two numbers are such that their highest common factor is $150$ and lowest common multiple is $600$. Find all the possible two numbers if all the numbers are at least $120$.

Math Question Bank