Practice Questions

Given two distinct prime numbers $p$ and $q$, formulate general expressions for their HCF (Highest Common Factor) and LCM (Least Common Multiple).

Calculate the HCF of 144 and 198 using the prime factorization method.

Express the number 4095 as a product of its prime factors.

The product of two numbers is 2160 and their HCF is 12. Calculate their LCM.

Name the type of decimal expansion that an irrational number has.

The prime factorization of a number is $2^3 \times 3 \times 5^2$. Without performing the full calculation of the number, justify whether it must end with the digit 0.

Justify why the expression $17 \times 19 \times 23 + 23$ represents a composite number without performing the full calculation.

If a number's prime factorization is $2^3 \times 3 \times 5^2$, list all its distinct prime factors.

Summarize the procedure to find the HCF and LCM of two positive integers using the prime factorization method.

Calculate the HCF and LCM of 42 and 72. Then, verify that HCF $\times$ LCM is equal to the product of the two numbers.

List the first five prime numbers and the first five composite numbers.

Describe the two main types of real numbers. For each type, provide a definition and list three different examples.

Define a composite number.

State the Fundamental Theorem of Arithmetic.

Given that $\sqrt{5}$ is an irrational number, demonstrate that $7 - 2\sqrt{5}$ is also an irrational number.

Evaluate whether the product of two irrational numbers is always irrational. Justify your conclusion with a suitable example.

Identify the relationship between the HCF and LCM of two positive integers, $a$ and $b$.

Explain why the number $6^n$, where $n$ is any natural number, can never end with the digit 0.

Explain the fundamental difference between Euclid's division algorithm and the Fundamental Theorem of Arithmetic.

Summarize the key steps in the proof that $\sqrt{3}$ is an irrational number, using the method of 'proof by contradiction'.

At a running track, three athletes, Aman, Bimal, and Charan, start running from the same point at the same time in the same direction. They complete one round in 24 seconds, 36 seconds, and 48 seconds, respectively. Calculate the time after which they will all meet again at the starting point for the first time. How many rounds will each have completed by then?

Create a proof to show that if $p$ is any prime number, then $\sqrt{p}$ is irrational. Your proof must follow the method of contradiction and clearly state the key theorem used.

Describe what it means for two integers to be 'coprime'. Provide one example of a pair of coprime numbers and one example of a pair that is not coprime.

Recall Theorem 1.2 from the text. State the theorem and provide a simple example to illustrate it.

Examine if the number $9^n$ can end with the digit 0 for any natural number $n$. Justify your answer.

Explain why the number $3 \times 5 \times 17 + 17$ is a composite number.

Using the prime factorization method, find the HCF and LCM of 40, 60, and 100.

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

Explain the meaning of the 'uniqueness' part of the Fundamental Theorem of Arithmetic. Use the number 360 to illustrate your explanation.

A milk vendor has three containers of milk with capacities 42 litres, 56 litres, and 63 litres. He wants to measure the milk from the three containers using a smaller can. What is the maximum capacity of the can that he can use to measure the milk from each container an exact number of times? How many times will he have to use the can for each container?

A student claims that for any natural number $n$, the number $12^n$ can end with the digit 0. Critique this claim using the Fundamental Theorem of Arithmetic.

Prove that $\sqrt{7}$ is an irrational number, given that 7 is a prime number.

In the standard proof that $\sqrt{3}$ is irrational, we begin by assuming that $\sqrt{3} = \frac{a}{b}$, where $a$ and $b$ are coprime. Justify why this 'coprime' condition is critical to the structure of the proof.

Two alarm clocks ring at regular intervals of 50 seconds and 48 seconds respectively. If they first beep together at 12:00 noon, formulate a method based on prime factorization to determine the time when they will next beep together.

Prove that $7 - 2\sqrt{3}$ is an irrational number, assuming it is given that $\sqrt{3}$ is irrational.

A sweetseller has 420 kaju barfis and 130 badam barfis. She wants to stack them in such a way that each stack has the same number of barfis, and the stacks take up the least area of the tray. Evaluate the number of barfis that should be placed in each stack for this purpose. Justify your method.

The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other number.

Find the largest number that divides 245 and 1029, leaving a remainder of 5 in each case.

Create a proof to show that for any positive integer $n$, the number $n^3 - n$ is divisible by 6.

Design a method using prime factorization to find the largest positive integer that will divide 398, 436, and 542 leaving remainders 7, 11, and 15, respectively. Execute your method.

For any two positive integers $a$ and $b$, prove that the product of the numbers is equal to the product of their HCF and LCM, i.e., $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$. Base your proof on the Fundamental Theorem of Arithmetic.

Calculate the smallest positive integer which when divided by 15, 20, and 30 leaves a remainder of 5 in each case.

Explain how the prime factorization of the denominator of a rational number determines whether its decimal expansion is terminating or non-terminating repeating.

An army contingent of 612 members is to march behind an army band of 48 members in a parade. The two groups are to march in the same number of columns. Design a procedure using the Fundamental Theorem of Arithmetic to find the maximum number of columns in which they can march. Formulate the final arrangement.