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PSLE

<p>$F_x$ is the set of positive integer factors of $x$. For example, $F_9 = \{1, 3, 9\}$.</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">List the elements of</div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-paragraph">$F_{24}$,</div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">$F_{18} \cap F_{12}$.</div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Show that&nbsp;$n(F_{16}) = 5$.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Find a possible value of $x$ such that</div><ol class="i-list"><li data-list-item-id="item-c-i" data-list-item-path="item-c/item-c-i"><div class="list-item-paragraph">$F_x \subseteq F_{32}$,</div></li><li data-list-item-id="item-c-ii" data-list-item-path="item-c/item-c-ii"><div class="list-item-paragraph">$n(F_x) = 3$.</div></li></ol></li></ol>
<p>Take $x$ to be a positive integer.</p>

$F_x$ is the set of positive integer factors of $x$. For example, $F_9 = \{1, 3, 9\}$.

Introduction to Set Notations

Representing Relationships between Sets

Intersection of Two Sets

Factors, Multiples and Divisibility of Integers

IP2U10 HOT Q4

Sets

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 $864$ 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">Written as a product of its prime factors, $720 = 2^{4} \times 3^{2} \times 5$. Find the smallest possible positive integer value of $n$ such $720n$ is a multiple of $864$.</div></li></ol>

Express $864$ as the product of its prime factors.

Written as a product of its prime factors, $720 = 2^{4} \times 3^{2} \times 5$. Find the smallest possible positive integer value of $n$ such $720n$ is a multiple of $864$.

Prime Factorisation of Positive Integers

IP1U1 SWII Q7

<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express $360$ as a product of its prime factors.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the smallest positive integer $k$ such that $360k$ is a multiple of $196$.</span></div></li></ol>

Express $360$ as a product of its prime factors.

Find the smallest positive integer $k$ such that $360k$ is a multiple of $196$.

IP1U1 SWI Q11

<p>Given that $r$ and $s$ are two different prime numbers, list all the factors of $rs^3$.</p>

Given that $r$ and $s$ are two different prime numbers, list all the factors of $rs^3$.

IP1U1 SWI Q9

<p>Given that $p$ and $q$ are prime numbers, determine if '$pq^2$ has only $3$ factors' is true or false. Explain your answer.</p>

Given that $p$ and $q$ are prime numbers, determine if '$pq^2$ has only $3$ factors' is true or false. Explain your answer.

IP1U1 SWI Q8

<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 $350$.

Perfect Squares and Cubes Using Prime Factors

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

<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>

Highest Common Factor and Its Applications

Lowest Common Multiple and Its Applications

Prime Factorisation and Number Properties of 3780

<p>Written as the product of prime factors $3500=2^{2}\times 5^{3}\times 7$.</p>
<ol class="abc-list"><li><div class="list-item-paragraph">Express $720$ as the product of its prime factors.</div></li><li><div class="list-item-paragraph">Hence write down</div><ol class="i-list"><li><div class="list-item-paragraph">the largest integer which is a factor of both $3500$ and $720$,</div></li><li><div class="list-item-paragraph">the smallest positive integer $k$ for which $720k$ is a multiple of $3500$.</div></li></ol></li></ol>

Written as the product of prime factors $3500=2^{2}\times 5^{3}\times 7$.

Prime Factorisation, HCF, and LCM Applications

<ol class="abc-list"><li><div class="list-item-paragraph">Use a factor tree to find the prime factorization of $630$, leaving your answer in index notation.</div></li><li><div class="list-item-paragraph">The prime factorization of a number is $2^{4}\times 3^{2}\times 5$. Hence, write down $2$ factors of the number which are greater than $100$.</div></li></ol>

Prime Factorisation and Factors of a Number

<p>The numbers $54$ and $468$, written as the product of their prime factors, are $54=2\times {{3}^{3}}$ and $468={{2}^{2}}\times {{3}^{2}}\times 13$. Find</p>
<ol class="abc-list"><li data-list-item-id="item-1760761620645-vcucpphy5" data-list-item-path="item-1760761620645-vcucpphy5"><div class="list-item-paragraph">the smallest positive integer, $m$ such that&nbsp;$468m$ is a perfect square,</div></li><li data-list-item-id="item-1760761621140-mrqmdyd59" data-list-item-path="item-1760761621140-mrqmdyd59"><div class="list-item-paragraph">the largest integer, which is a factor of both&nbsp;$54$ and $468$,</div></li><li data-list-item-id="item-1760761621859-pwrqeuuc1" data-list-item-path="item-1760761621859-pwrqeuuc1"><div class="list-item-paragraph">the smallest positive integer, $n$ such that&nbsp;$54n$ is a multiple of $468$.</div></li></ol>

The numbers $54$ and $468$, written as the product of their prime factors, are $54=2\times {{3}^{3}}$ and $468={{2}^{2}}\times {{3}^{2}}\times 13$. Find

Applications of Prime Factorisation in Number Problems

<ol class="i-list"><li><div class="list-item-paragraph">Find the smallest positive integer $n$ for which $\sqrt{136n}$ is a whole number.</div></li><li><div class="list-item-paragraph">Find the smallest positive integer $k$ for which $240k$ is a multiple of $7056$.</div></li><li><div class="list-item-paragraph">When $180$ is multiplied by a whole number $k$, the product is a perfect cube. Suggest two possible values of $k$.</div></li></ol>

Higher Order Thinking Questions

<p>If&nbsp;$p$, $q$ and $r$ are whole numbers such that&nbsp;$p\times q\times r=142$. Given that&nbsp;$r$ is an even number, find all possible values of $p+q$. Explain your answer.</p>

If $p$, $q$ and $r$ are whole numbers such that $p\times q\times r=142$. Given that $r$ is an even number, find all possible values of $p+q$. Explain your answer.

Finding Unknown Factors from a Product Constraint

<p>If&nbsp;$p$, $q$ and $r$ are whole numbers such that&nbsp;$p\times q\times r=34$. Given that&nbsp;$r$ is an even number, find all possible values of $p+q$. Explain your answer.</p>

If $p$, $q$ and $r$ are whole numbers such that $p\times q\times r=34$. Given that $r$ is an even number, find all possible values of $p+q$. Explain your answer.

<p>The product of the ages of Mrs Lee and her three children is $5180$. If none of her children differ by more than $3$ years in age, find the average of the ages of Mrs Lee and her children.</p>
<p>For the intended age model, take all ages as positive whole numbers of years, with Mrs Lee older than her children and younger than 100.</p>

The product of the ages of Mrs Lee and her three children is $5180$. If none of her children differ by more than $3$ years in age, find the average of the ages of Mrs Lee and her children.

For the intended age model, take all ages as positive whole numbers of years, with Mrs Lee older than her children and younger than 100.

Mean of Ungrouped Data

Average Age Problem with Constraints on Age Differences

Statistics

<ol class="abc-list"><li data-list-item-id="item-1758616108085-j1v1h3ffa" data-list-item-path="item-1758616108085-j1v1h3ffa"><ol class="i-list"><li data-list-item-id="subitem-1758616109517-0ilsci5an" data-list-item-path="item-1758616108085-j1v1h3ffa/subitem-1758616109517-0ilsci5an"><div class="list-item-paragraph">Express $594$ as a product of its prime factors.</div></li><li data-list-item-id="subitem-1758616109900-2k6sh0j0u" data-list-item-path="item-1758616108085-j1v1h3ffa/subitem-1758616109900-2k6sh0j0u"><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-1758616117977-z7i02uk7r" data-list-item-path="item-1758616117977-z7i02uk7r"><div class="list-item-paragraph">Megan is playing with $594$ cubes. Megan uses all $594$ cubes to make a cuboid. 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-1758616118990-5cgufu51b" data-list-item-path="item-1758616118990-5cgufu51b"><div class="list-item-paragraph">Each cube has a volume of $2$cm$^{3}$. 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>

Megan is playing with $594$ cubes. Megan uses all $594$ cubes to make a cuboid. 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.

Each cube has a volume of $2$cm$^{3}$. Calculate the maximum number of cubes that Megan can fit into a box of dimensions $8$cm by $10$cm by $10$cm.

IP1U1 HOT Q1

<ol class="abc-list"><li data-list-item-id="item-1758604507701-zz8ozovas" data-list-item-path="item-1758604507701-zz8ozovas"><div class="list-item-paragraph">Express $3780$ as the product of its prime factors.</div></li><li data-list-item-id="item-1758604508369-imx4usvw9" data-list-item-path="item-1758604508369-imx4usvw9"><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-1758604510214-jxc1jhbvk" data-list-item-path="item-1758604510214-jxc1jhbvk"><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-1758604510866-b8cni9mnc" data-list-item-path="item-1758604510866-b8cni9mnc"><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>

Express $3780$ as the product of its prime factors.

Find the smallest positive integer $p$ such that $3780p$ is a perfect cube.

Find the smallest positive integer $q$ such that $3780q$ is a multiple of $350$.

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 $k$.

Math Question Bank