Vardaan Learning Institute

Express 140 as a product of its prime factors.

Express 156 as a product of its prime factors.

Express 3825 as a product of its prime factors.

Express 5005 as a product of its prime factors.

Express 7429 as a product of its prime factors.

Find the prime factorization of 945 and determine the sum of the exponents of its prime factors.

State the Fundamental Theorem of Arithmetic. Using it, check whether $6^n$ can end with the digit 0 for any natural number $n$.

Check whether $4^n$ can end with the digit 0 for any natural number $n$. Justify your answer.

Explain why $7 \times 11 \times 13 + 13$ is a composite number.

Explain why $7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 + 5$ is a composite number.

If $n$ is any natural number, then $6^n - 5^n$ always ends with which digit? Justify your answer.

Find the LCM and HCF of 26 and 91 by prime factorization method.

Find the LCM and HCF of 510 and 92 by prime factorization.

Find the LCM and HCF of 336 and 54 using the prime factorization method.

Find the HCF and LCM of 12, 15 and 21 by applying the prime factorization method.

Find the HCF and LCM of 8, 9 and 25 by prime factorization.

Given that $\text{HCF}(306, 657) = 9$, find $\text{LCM}(306, 657)$.

The LCM of two numbers is 182 and their HCF is 13. If one of the numbers is 26, find the other number.

If the HCF of 65 and 117 is expressible in the form $65m - 117$, find the value of $m$.

If the HCF of 408 and 1032 is expressible in the form $1032m - 408 \times 5$, find $m$.

Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.

If $p$ and $q$ are two distinct prime numbers, what is their HCF and LCM?

The LCM of two co-prime numbers is 4875. If one of the numbers is 75, find the other.

Write the smallest number which is divisible by both 306 and 657.

If $a = x^3 y^2$ and $b = x y^3$, where $x$ and $y$ are prime numbers, find the HCF and LCM of $a$ and $b$.

If $p = a^2 b^3$ and $q = a^3 b$, find $\text{HCF}(p, q) \times \text{LCM}(p, q)$.

Three numbers are in the ratio $2:3:4$ and their HCF is 12. Find the LCM of the numbers.

There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting point?

Three bells toll at intervals of 9, 12, and 15 minutes respectively. If they start tolling together, after what time will they next toll together?

Four traffic lights on different road crossings change after every 24 seconds, 36 seconds, 48 seconds, and 72 seconds respectively. If they change simultaneously at 8:00 AM, at what time will they change together again?

In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that all can cover the same distance in complete steps?

Find the least number which when divided by 12, 15, 20 and 54 leaves a remainder of 8 in each case.

Find the smallest number which when divided by 28 and 32 leaves remainders 8 and 12 respectively.

Two tankers contain 850 litres and 680 litres of kerosene oil respectively. Find the maximum capacity of a container which can measure the kerosene oil of both the tankers when used an exact number of times.

A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of barfis that can be placed in each stack for this purpose?

Three sets of English, Hindi and Mathematics books have to be stacked so that all books are stored subject-wise and the height of each stack is the same. The number of English books is 96, Hindi books is 240 and Mathematics books is 336. Assuming the books are of the same thickness, determine the number of stacks of English, Hindi and Mathematics books.

Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.

Find the largest number which divides 2053 and 967 and leaves remainders of 5 and 7 respectively.

Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively.

Find the greatest number of 6 digits exactly divisible by 24, 15 and 36.

Find the smallest number of 4 digits which is exactly divisible by 12, 15, 20 and 35.

A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of a third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?

There are 156, 208 and 260 students in groups A, B and C respectively. Buses are to be hired to take them for a trip. Find the minimum number of buses to be hired if the same number of students should be accommodated in each bus and a bus should not have students from different groups.

The length, breadth and height of a room are 8m 25cm, 6m 75cm and 4m 50cm respectively. Determine the longest tape which can measure the three dimensions of the room exactly.

Two alarm clocks ring their alarms at regular intervals of 50 seconds and 48 seconds. If they first beep together at 12 noon, at what time will they beep together again for the first time?

Find the least number which when divided by 35, 56 and 91 leaves the same remainder 7 in each case.

Determine the number nearest to 110000 but greater than 100000 which is exactly divisible by 8, 15 and 21.

A rectangular courtyard is $18\text{ m } 72\text{ cm}$ long and $13\text{ m } 20\text{ cm}$ broad. It is to be paved with square tiles of the same size. Find the least possible number of such tiles.

Prove that $\sqrt{2}$ is an irrational number.

Prove that $\sqrt{3}$ is an irrational number.

Prove that $\sqrt{5}$ is an irrational number.

Prove that $\sqrt{7}$ is an irrational number.

Prove that $\sqrt{11}$ is an irrational number.

Prove that $3 + 2\sqrt{5}$ is irrational, given that $\sqrt{5}$ is irrational.

Prove that $\dfrac{1}{\sqrt{2}}$ is irrational.

Prove that $7\sqrt{5}$ is irrational, given that $\sqrt{5}$ is irrational.

Prove that $6 + \sqrt{2}$ is irrational, given that $\sqrt{2}$ is irrational.

Prove that $5 - \sqrt{3}$ is an irrational number.

Prove that $\sqrt{2} + \sqrt{3}$ is irrational.

Is the product of a non-zero rational number and an irrational number always irrational? Justify with a formal proof.

Assertion (A): The HCF of two numbers is 5 and their product is 150. Then their LCM is 30.
Reason (R): For any two positive integers $a$ and $b$, $\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is NOT the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

Assertion (A): $6^n$ ends with the digit 0, where $n$ is a natural number.
Reason (R): Any number ending with digit 0 must have both 2 and 5 as its prime factors.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is NOT the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

Assertion (A): $\sqrt{2}$ is an irrational number.
Reason (R): If $p$ is a prime number and $p$ divides $a^2$, then $p$ divides $a$ (where $a$ is a positive integer).

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is NOT the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

Assertion (A): The LCM of 12, 21, and 15 is 420.
Reason (R): The LCM is the product of the lowest powers of common prime factors.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is NOT the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

Assertion (A): The number $5 \times 11 \times 13 + 13$ is a composite number.
Reason (R): A composite number has at least one factor other than 1 and itself.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is NOT the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

Assertion (A): Two numbers can have 12 as their HCF and 350 as their LCM.
Reason (R): LCM of two numbers is always a multiple of their HCF.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is NOT the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.

Assertion (A): $\dfrac{p}{q}$ is a terminating decimal if $q$ is of the form $2^m \times 5^n$, where $m$ and $n$ are non-negative integers.
Reason (R): All rational numbers are terminating decimals.

1. Both A and R are true and R is the correct explanation of A.
2. Both A and R are true but R is NOT the correct explanation of A.
3. A is true but R is false.
4. A is false but R is true.