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PSLE Foundation Mathematics

<p>Ellen and Felicia jogged around a circular track. They started at the starting line at the same time. Ellen took $64$ seconds and Felicia took $80$ seconds to complete one round.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Assuming that the two girls continued running at their respective speed, after how long would they meet again at the starting line? Give your answer in minutes and seconds.</span></div></li><li><div class="list-item-content"><span class="list-item-text">How many rounds would each girl have jogged by then?</span></div></li></ol>

Ellen and Felicia jogged around a circular track. They started at the starting line at the same time. Ellen took $64$ seconds and Felicia took $80$ seconds to complete one round.

Assuming that the two girls continued running at their respective speed, after how long would they meet again at the starting line? Give your answer in minutes and seconds.

How many rounds would each girl have jogged by then?

Lowest Common Multiple and Its Applications

IP1U1 SWIII Q10

Primes, HCF and LCM

<p>Three bus services are scheduled to leave Ang Mo Kio bus interchange at regular intervals of $9$ minutes, $12$ minutes and $16$ minutes respectively. Given that the three bus services leave the bus interchange together at $5.30$ am, determine the time when the three bus services next leave the bus interchange together.</p>

Three bus services are scheduled to leave Ang Mo Kio bus interchange at regular intervals of $9$ minutes, $12$ minutes and $16$ minutes respectively. Given that the three bus services leave the bus interchange together at $5.30$ am, determine the time when the three bus services next leave the bus interchange together.

IP1U1 SWIII Q7

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the lowest common multiple (LCM) of $15$, $32$ and $48$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Due to a safety regulation, three warning alarms will sound off a 'beep' every $15$ minutes, $32$ minutes and $48$ minutes respectively. Given that they sounded off together at $6.00$ a.m. on one particular morning, when is the next time that all the three alarms will beep together again?</span></div></li></ol>

Find the lowest common multiple (LCM) of $15$, $32$ and $48$.

Due to a safety regulation, three warning alarms will sound off a 'beep' every $15$ minutes, $32$ minutes and $48$ minutes respectively. Given that they sounded off together at $6.00$ a.m. on one particular morning, when is the next time that all the three alarms will beep together again?

IP1U1 SWIII Q3

<p>Three comets pass through our solar system every $42$ years, $60$ years and $330$ years respectively. The last time they passed through our solar system together was in the year $1641$. When will they next pass through our solar system together?</p>

Three comets pass through our solar system every $42$ years, $60$ years and $330$ years respectively. The last time they passed through our solar system together was in the year $1641$. When will they next pass through our solar system together?

IP1U1 SWIII Q1

<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 $r$.

Highest Common Factor 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 LCM of $6$, $15$ and a number, $x$ is $660$. Find the smallest possible positive integer value of $x$.</p>

The LCM of $6$, $15$ and a number, $x$ is $660$. Find the smallest possible positive integer value of $x$.

IP1U1 SWII Q16

<ol class="i-list"><li><div class="list-item-paragraph">Find the lowest common multiple of the following three terms.</div><div class="list-item-paragraph" style="text-align: center">${a^{2} \times b \times c^{3}}$, ${a \times b^{2} \times c \times d^{3}}$ and ${a \times b^{3} \times c^{2} \times d^{2}}$</div></li><li><div class="list-item-content"><span class="list-item-text">Find the LCM of $m^{2}np$ and $mn^{3}$.</span></div></li></ol>

Find the LCM of $m^{2}np$ and $mn^{3}$.

IP1U1 SWII Q15

<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

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

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

IP1U1 SWII Q6

<p>Find the lowest common multiple (LCM) of the following numbers, $48$, $72$ and $120$.</p>

Find the lowest common multiple (LCM) of the following numbers, $48$, $72$ and $120$.

IP1U1 SWII Q5

<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.

Prime Factorisation of Positive Integers

IP1 Class Test (May 2025) Q3

<p>For the following number sequence: \[\frac{1}{2}, \frac{3}{7}, \frac{5}{12}, \frac{7}{17}, \frac{9}{22}, \dots\]</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Write down the next two terms of the sequence.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find a formula for the $n^{th}$ term of the sequence in terms of $n$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the $16^{th}$ term.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the lowest common multiple of the denominators of the $7^{th}$ and the $11^{th}$ term. Hence, find the difference between the $7^{th}$ and the $11^{th}$ term.</span></div></li></ol>

For the following number sequence: $$\frac{1}{2}, \frac{3}{7}, \frac{5}{12}, \frac{7}{17}, \frac{9}{22}, \dots$$

Write down the next two terms of the sequence.

Find a formula for the $n^{th}$ term of the sequence in terms of $n$.

Find the lowest common multiple of the denominators of the $7^{th}$ and the $11^{th}$ term. Hence, find the difference between the $7^{th}$ and the $11^{th}$ term.

General Term of a Sequence

 Number Patterns

<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 $4, 7, 12, 19, 28$.

<p>Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = 56 - 4n$, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the $3^{rd}$ term,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the difference between the $15^{th}$ and the $5^{th}$ term,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the lowest common multiple of the $10^{th}$ and the $11^{th}$ term.</span></div></li></ol>

Given that the $n^{th}$ term, $T_n$, of a sequence is $T_n = 56 - 4n$, find

the difference between the $15^{th}$ and the $5^{th}$ term,

the lowest common multiple of the $10^{th}$ and the $11^{th}$ term.

<p>When&nbsp;$x$ marbles are packed into bags of $10$, $22$ or $25$, there always remain $3$ marbles not packed into any of the bags.</p>
<p>What is the smallest possible value of $x$?</p>
<p>In each packing arrangement, at least one full bag is packed.</p>

When $x$ marbles are packed into bags of $10$, $22$ or $25$, there always remain $3$ marbles not packed into any of the bags.

What is the smallest possible value of $x$?

In each packing arrangement, at least one full bag is packed.

 Lowest Common Multiple and Its Applications

<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 $120$.

Lowest Common Multiple and Highest Common Factor and Their Applications

<p>Given that the lowest common multiple of $30$, $126$ and $p$ is $4410$, where $p$ is a whole number.</p>
<p>Find the smallest value of $p$.</p>

Given that the lowest common multiple of $30$, $126$ and $p$ is $4410$, where $p$ is a whole number.

<p>Cakes cost SGD 5.60 per slice, muffins cost SGD 1.60 each and cookies cost SGD 0.90 each. The bakery wants equal daily revenue from each type of food.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">What is the minimum amount of money he could earn for each item?</div></li><li data-list-item-id="item-1767528715914-u17yittq5" data-list-item-path="item-1767528715914-u17yittq5"><div class="list-item-paragraph">How many muffins did he sell?</div></li></ol>
<p>Assume at least one of each type of food is sold. Find the minimum positive equal revenue from each type.</p>

Cakes cost SGD 5.60 per slice, muffins cost SGD 1.60 each and cookies cost SGD 0.90 each. The bakery wants equal daily revenue from each type of food.

What is the minimum amount of money he could earn for each item?

Assume at least one of each type of food is sold. Find the minimum positive equal revenue from each type.

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