A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB HL Math

IB SL Math

IP Math

<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>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.

Highest Common Factor and Its Applications

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?

IP1U1 SWIII Q8

<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

<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><div class="list-item-content"><span class="list-item-text">Find the total number of pieces of ribbons that Mary has cut.</span></div></li><li><div class="list-item-content"><span class="list-item-text">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.</span></div></li></ol>

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.

IP1U1 SWIII Q6

<p>The Singapore Flyer is a giant observation wheel with seating capacity of \(784\) passengers.</p>
<p>The number of passengers that each capsule can carry is equal to the total number of capsules on the Singapore Flyer.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the prime factorisation of \(784\) and express your answer in index notation.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find the total number of capsules on the Singapore Flyer.</span></div></li></ol>

The Singapore Flyer is a giant observation wheel with seating capacity of \(784\) passengers.

The number of passengers that each capsule can carry is equal to the total number of capsules on the Singapore Flyer.

Find the prime factorisation of \(784\) and express your answer in index notation.

Hence, find the total number of capsules on the Singapore Flyer.

Prime Factorisation and Its Applications

IP1U1 SWIII Q5

<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

<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>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>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 \(r\).

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 value of \(x\).</p>

The LCM of \(6\), \(15\) and a number, \(x\) is \(660\). Find the smallest possible 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 lowest common multiple of the following three terms. \[\(a^{2} \times b \times c^{3}\), \(a \times b^{2} \times c \times d^{3}\) and \(a \times b^{3} \times c^{2} \times d^{2}\)\]

Find the LCM of \(m^{2}np\) and \(mn^{3}\).

IP1U1 SWII Q15

<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 \(\sqrt[3]{1728x^9}\).

IP1U1 SWII Q14

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Express \(2744\) as the product of its prime factors. Give your answer in index notation.</span></div></li><li><div class="list-item-paragraph">State the smallest integer value of \(n\), for \(\frac{2744}{n}\) to be a perfect square.</div></li><li><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.

State the smallest integer value of \(n\), for \[\frac{2744}{n}\] to be a perfect square.

Use your answer from part (a) to find the value of \(\sqrt[3]{2744}\).

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.

IP1U1 SWII Q12

<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the prime factorization of \(2772\), expressing your answer in index notation.</span></div></li><li><div class="list-item-content"><span class="list-item-text">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 \(4200\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the smallest integer \(k\) such that \((2772 \times 4200k)\) is a square number.</span></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 integer \(k\) such that \((2772 \times 4200k)\) is a square number.

IP1U1 SWII Q11

<p>Expressed as the product of its prime numbers, \(3375=3^{3} \times 5^{3}\) and \(32400=2^{4} \times 3^{4} \times 5^{2}\).</p>
<p>Using these results,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">find in prime factorization form, the highest common factor of \(3375\) and \(32400\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">find the value of \(\sqrt{32400} \div \sqrt[3]{3375}\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">find the smallest positive integer \(k\), such that \(3375k\) is a perfect square.</span></div></li></ol>

Expressed as the product of its prime numbers, \(3375=3^{3} \times 5^{3}\) 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><div class="list-item-content"><span class="list-item-text">Express \(2744\) as a product of its prime factors, giving your answer in index notation.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, find \(\sqrt[3]{2744}\) without the use of a calculator.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that \(2744k\) is a perfect square, write down the smallest possible integer value of \(k\).</span></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 integer value of \(k\).

Math Question Bank