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PSLE

PSLE Foundation Mathematics

<p>The number $1728$ written as a product of its prime factors is $1728 = 2^{6} \times 3^{3}$.</p>
<p>Using only the above result and showing your working clearly, find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the smallest positive integer value of $k$ for which $1728k$ is a square number,</span></div></li><li><div class="list-item-content"><span class="list-item-text">an expression in terms of $x$ for $\sqrt[3]{1728x^9}$.</span></div></li></ol>

The number $1728$ written as a product of its prime factors is $1728 = 2^{6} \times 3^{3}$.

Using only the above result and showing your working clearly, find

the smallest positive integer value of $k$ for which $1728k$ is a square number,

an expression in terms of $x$ for $\sqrt[3]{1728x^9}$.

Perfect Squares and Cubes Using Prime Factors

Square and Cube Roots Using Prime Factorisation

IP1U1 SWII Q14

Primes, HCF and LCM

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Express $2744$ as the product of its prime factors. Give 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">State the smallest positive integer value of $n$, for&nbsp;$\frac{2744}{n}$ to be a perfect square.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Use your answer from part <strong>(a)</strong> to find the value of $\sqrt[3]{2744}$.</div></li></ol>

Express $2744$ as the product of its prime factors. Give your answer in index notation.

Prime Factorisation of Positive Integers

IP1U1 SWII Q13

<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.

Highest Common Factor and Its Applications

Lowest Common Multiple and Its Applications

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.

IP1U1 SWII Q10

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Express $2744$ as a product of its prime factors, giving 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">Hence, find $\sqrt[3]{2744}$ without the use of a calculator.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Given that&nbsp;$2744k$ is a perfect square, write down the smallest possible positive integer value of $k$.</div></li></ol>

Express $2744$ as a product of its prime factors, giving your answer in index notation.

Hence, find $\sqrt[3]{2744}$ without the use of a calculator.

Given that $2744k$ is a perfect square, write down the smallest possible positive integer value of $k$.

IP1U1 SWII Q9

<p>Given that&nbsp;$\frac{2835}{3m}$ is a perfect cube, find the smallest positive integer value of $m$.</p>

Given that $\frac{2835}{3m}$ is a perfect cube, find the smallest positive integer value of $m$.

IP1U1 SWII Q8

<ol class="i-list"><li data-list-item-id="item-2-i" data-list-item-path="item-2-i"><div class="list-item-paragraph">Express $3528$ as the product of its prime factors.</div></li><li data-list-item-id="item-2-ii" data-list-item-path="item-2-ii"><div class="list-item-paragraph">Find the smallest positive integer $k$, such that&nbsp;$3528k$ is a cube number.</div></li></ol>

Express $3528$ as the product of its prime factors.

Find the smallest positive integer $k$, such that $3528k$ is a cube number.

IP1U1 SWI Q15

<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">Express $10692$ as the product of its prime factors.</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">Hence, find the smallest positive integer value of $k$ such that&nbsp;$\sqrt{\frac{10692}{k}}$ is an integer.</div></li></ol>

Express $10692$ as the product of its prime factors.

Hence, find the smallest positive integer value of $k$ such that $\sqrt{\frac{10692}{k}}$ is an integer.

IP1U1 SWI Q14

<p>Given that $392040 = 2^3 \times 3^4 \times 5 \times 11^2$ when expressed as a product of its prime factors, find</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the smallest positive integer $n$ such that $392040\sqrt{n}$ is a perfect square, and</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the largest positive integer $m$ such that $\frac{392040}{m}$ is a perfect cube.</span><span class="inline-marks">[1]</span></div></li></ol>

Given that $392040 = 2^3 \times 3^4 \times 5 \times 11^2$ when expressed as a product of its prime factors, find

the smallest positive integer $n$ such that $392040\sqrt{n}$ is a perfect square, and

the largest positive integer $m$ such that $\frac{392040}{m}$ is a perfect cube.

IP1 Class Test (May 2025) Q5

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Express $272\ 250$ as the product of its prime factors.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the smallest positive integer $k$ such that&nbsp;$\sqrt{\frac{272\ 250}{k}}$ is an integer.</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 $q$ such that&nbsp;$272\ 250q$ is a multiple of $350$.</div></li></ol>

Express $272\ 250$ as the product of its prime factors.

Find the smallest positive integer $k$ such that $\sqrt{\frac{272\ 250}{k}}$ is an integer.

Find the smallest positive integer $q$ such that $272\ 250q$ is a multiple of $350$.

Factors, Multiples and Divisibility of Integers

Prime Factorisation and Its Applications

<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Express $11\ 250$ as the product of its prime factors.</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Find the smallest positive integer $k$ such that&nbsp;$\sqrt[3]{11\ 250k}$ is an integer.</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 $q$ such that&nbsp;$11\ 250q$ is a multiple of $450$.</div></li></ol>

Express $11\ 250$ as the product of its prime factors.

Find the smallest positive integer $k$ such that $\sqrt[3]{11\ 250k}$ is an integer.

Find the smallest positive integer $q$ such that $11\ 250q$ is a multiple of $450$.

 Prime Factorisation and Its Applications

<p>The numbers $1680$ and $14\ 850$, expressed as the product of their prime factors, are $1680 = 2^4 \times 3 \times 5 \times 7$ and $14\ 850 = 2 \times 3^3 \times 5^2 \times 11$.</p>
<p>Hence, find</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">the largest integer which is a factor of $1680$ and $14\ 850$,</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">the smallest positive integer $k$, such that&nbsp;$14\ 850k$ is a perfect square,</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">the smallest positive integer $m$, such that&nbsp;$1680 \times 14\ 850 \times m$ is a perfect cube.</div></li></ol>

The numbers $1680$ and $14\ 850$, expressed as the product of their prime factors, are $1680 = 2^4 \times 3 \times 5 \times 7$ and $14\ 850 = 2 \times 3^3 \times 5^2 \times 11$.

the largest integer which is a factor of $1680$ and $14\ 850$,

the smallest positive integer $k$, such that $14\ 850k$ is a perfect square,

the smallest positive integer $m$, such that $1680 \times 14\ 850 \times m$ is a perfect cube.

<p>Written as a product of its prime factors, $50\ 960 = 2^4 \times 5 \times 7^2 \times 13$.</p>
<p>Find the smallest positive integer $k$ such that</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">$\frac{50\ 960}{k}$ is a perfect cube,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$50\ 960k$ is a perfect square.</span></div></li></ol>

Written as a product of its prime factors, $50\ 960 = 2^4 \times 5 \times 7^2 \times 13$.

Find the smallest positive integer $k$ such that

<ol class="abc-list"><li><div class="list-item-paragraph">Express $10692$ as the product of its prime factors.
</div></li><li><div class="list-item-paragraph">Hence, find the smallest positive integer value of $k$ such that $\sqrt{\frac{10692}{k}}$ is an integer.</div></li></ol>

[EPQ Vol 1A] Question 5

<ol class="abc-list"><li data-list-item-id="item-1763492053760-bipr7jpgp" data-list-item-path="item-1763492053760-bipr7jpgp"><ol class="i-list"><li data-list-item-id="subitem-1763492055380-elhdr4h1n" data-list-item-path="item-1763492053760-bipr7jpgp/subitem-1763492055380-elhdr4h1n"><div class="list-item-paragraph">Express $594$ as a product of its prime factors.</div></li><li data-list-item-id="subitem-1763492056597-1bcyh6580" data-list-item-path="item-1763492053760-bipr7jpgp/subitem-1763492056597-1bcyh6580"><div class="list-item-paragraph">Find the smallest positive integer $k$ such that&nbsp;$\frac{594}{\sqrt{k}}$ is a perfect cube.</div></li></ol></li><li data-list-item-id="item-1763492057878-hjhww496t" data-list-item-path="item-1763492057878-hjhww496t"><div class="list-item-paragraph">Megan is playing with $594$ cubes. Megan uses all $594$ cubes to make a cuboid.<br></div><div class="list-item-paragraph">Each of the sides of the cuboid is made up of more than $3$ cubes. Find the number of cubes on each side of the cuboid.</div></li><li data-list-item-id="item-1763492141249-xrgarlb7z" data-list-item-path="item-1763492141249-xrgarlb7z"><div class="list-item-paragraph">Each cube has a volume of $2$ cm³. Calculate the maximum number of cubes that Megan can fit into a box of dimensions $8$ cm by $10$ cm by $10$ cm.</div></li></ol>
<p>For part (c), arrange the cubes in straight rows and layers with their edges parallel to the box edges.</p>

For part (c), arrange the cubes in straight rows and layers with their edges parallel to the box edges.

Volume and Surface Area of Prisms

Packing Cubes and Cuboids

Prime Factorisation and Cube Problems Involving 594

<ol class="abc-list"><li data-list-item-id="item-1763491091542-vu3jmgec5" data-list-item-path="item-1763491091542-vu3jmgec5"><div class="list-item-paragraph">Express $3780$ as the product of its prime factors.</div></li><li data-list-item-id="item-1763491092497-x9csxkqju" data-list-item-path="item-1763491092497-x9csxkqju"><div class="list-item-paragraph">Find the smallest positive integer $p$ such that&nbsp;$3780p$ is a perfect cube.</div></li><li data-list-item-id="item-1763491093516-g5sv4sj2d" data-list-item-path="item-1763491093516-g5sv4sj2d"><div class="list-item-paragraph">Find the smallest positive integer $q$ such that&nbsp;$3780q$ is a multiple of $350$.</div></li><li data-list-item-id="item-1763491094118-8koexi6gd" data-list-item-path="item-1763491094118-8koexi6gd"><div class="list-item-paragraph">The lowest common multiple of $3780$ and integer $k$ is $7560$. The highest common factor of $3780$ and integer $k$ is $60$. Find the smallest possible positive integer value of integer $k$.</div></li></ol>

Prime Factorisation and Number Properties of 3780

<ol class="abc-list"><li><div class="list-item-paragraph">Express $14520$ as a product of prime factors.</div></li><li><div class="list-item-paragraph">Given that $14520k$ is a perfect square, write down the smallest positive integer $k$.</div></li><li><div class="list-item-paragraph">Find the largest integer which is a factor of $14520$ and $4400$.</div></li></ol>

Prime Factorisation, Perfect Squares, and Highest Common Factor

<ol class="abc-list"><li data-list-item-id="item-1763490717016-s92sd7g3x" data-list-item-path="item-1763490717016-s92sd7g3x"><div class="list-item-paragraph">Express $2744$ as a product of its prime factors, giving your answer in index notation.</div></li><li data-list-item-id="item-1763490717729-k6fh50a5w" data-list-item-path="item-1763490717729-k6fh50a5w"><div class="list-item-paragraph">Hence, find $\sqrt[3]{2744}$ without the use of a calculator.</div></li><li data-list-item-id="item-1763490718455-c5ctwrsvj" data-list-item-path="item-1763490718455-c5ctwrsvj"><div class="list-item-paragraph">Given that&nbsp;$2744k$ is a perfect square, write down the smallest possible positive integer value of $k$.</div></li></ol>

Prime Factorisation, Cube Roots, and Perfect Square Conditions

<ol class="abc-list"><li data-list-item-id="item-1763490523847-2fga4cy3d" data-list-item-path="item-1763490523847-2fga4cy3d"><div class="list-item-paragraph">Given&nbsp;$3780=2^{2}\times 3^{a}\times 5\times 7^{b}$ when expressed as a product of its prime factors. Write down the values of $a$ and of $b$.</div></li><li data-list-item-id="item-1763490524875-hwf8kmzdo" data-list-item-path="item-1763490524875-hwf8kmzdo"><div class="list-item-paragraph">Hence, find the smallest positive integer value of $k$, when&nbsp;$3780k$ is a perfect square.</div></li></ol>

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