Real Numbers — Class 10 Maths PYQs

If $3825 = 3^x \times 5^y \times 17^z$, then the value of $x + y - 2z$ is:

If the prime factorisation of $2520$ is $2^3 \times 3^a \times b \times 7$, then the value of $a + 2b$ is:

The total number of factors of the square of a prime number is :

Which of the following is a rational number between $\sqrt{3}$ and $\sqrt{5}$?

The sum of the exponents of prime factors in the prime factorisation of $4004$ is:

If $1080 = 2^P \times 3^q \times 5$, then $(p - q)$ is equal to :

If $a^b = 32$, where 'a' and 'b' are positive integers, then the value of $b^{ab}$ is :

Assertion (A) : The prime numbers which divide $36$ also divide $6$.
Reason (R) : Any number which divides $p^2$ also divides $p$.

If 'p' and 'q' are natural numbers and 'p' is the multiple of 'q', then what is the HCF of 'p' and 'q'?

The ratio of HCF to LCM of the least composite number and the least prime number is :

Two positive integers $m$ and $n$ are expressed as $m = p^5q^2$ and $n = p^3q^4$, where $p$ and $q$ are prime numbers. The LCM of $m$ and $n$ is :

If $a = 2^4 \times 3^3$, $b = 2^3 \times 3^2 \times 5$, $c = 3^n \times 5^2$ and LCM $(a, b, c) = (5^2 \times 3^4 \times 2^4)$, then $n$ is :

If two positive integers $p$ and $q$ can be expressed as $p = 18 a^2b^4$ and $q = 20 a^3b^2$, where $a$ and $b$ are prime numbers, then LCM $(p, q)$ is :

If the HCF $(2520, 6600) = 40$ and LCM $(2520, 6600) = 252 \times k$, then the value of $k$ is

If $a = 2^2 \times 3^x$, $b = 2^2 \times 3 \times 5$, $c = 2^2 \times 3 \times 7$ and LCM $(a, b, c) = 3780$, then $x$ is equal to

The HCF of two numbers $65$ and $104$ is $13$. If LCM of $65$ and $104$ is $40x$, then the value of $x$ is:

The LCM of three numbers $28, 44, 132$ is:

If the product of two co-prime numbers is $553$, then their HCF is :

The LCM of $24, 36$ and $60$ in terms of their prime factors is :

The ratio of the HCF to the LCM of $7, 21$ and $28$ is :

If $\text{HCF}(98, 28) = m$ and $\text{LCM}(98, 28) = n$, then the value of $n - 7m$ is:

The HCF of $40$, $110$ and $360$ is :

If $x$ is the LCM of $4, 6, 8$ and $y$ is the LCM of $3, 5, 7$ and $p$ is the LCM of $x$ and $y$, then which of the following is true ?

If $x = ab^3$ and $y = a^3b$, where $a$ and $b$ are prime numbers, then [HCF $(x, y)$ – LCM $(x, y)$] is equal to :

Let $x = a^2 b^3 c^n$ and $y = a^3 b^m c^2$, where $a, b, c$ are prime numbers. If LCM of $x$ and $y$ is $a^3 b^4 c^3$, then the value of $m+n$ is

Let $a = p^2 q^3 r^n$ and $b = p^3 q^m r^2$, where $p, q, r$ are prime numbers. If LCM of $a$ and $b$ is $p^3 q^4 r^3$, then the value of $3n - 2m$ is

Let $p = x^2 y^3 z^n$ and $q = x^3 y^m z^2$, where $x, y, z$ are prime numbers. If LCM $(p, q) = x^3 y^4 z^3$, then the value of $(2m + 3n)$ is

Assertion (A): For any two prime numbers $p$ and $q$, their HCF is 1 and LCM is $p+q$. Reason (R): For any two natural numbers, HCF $\times$ LCM = product of numbers.

Assertion (A): For two prime numbers $x$ and $y$ ($x < y$), $HCF(x, y) = x$ and $LCM(x, y) = y$. Reason (R): $HCF(x, y) \leq LCM(x, y)$, where $x, y$ are any two natural numbers.

Assertion (A) : For two odd prime numbers $x$ and $y$, $(x \neq y)$, $LCM(2x, 4y) = 4xy$. Reason (R) : $LCM(x, y)$ is a multiple of $HCF(x, y)$.

The HCF and the LCM of $14$, $21$ and $77$ respectively are

The HCF of $960$ and $432$ is:

The LCM of $960$ and $240$ is:

There are two sections A and B of Grade X. There are $28$ students in Section A and $30$ students in Section B. What is the minimum number of books you will acquire for the class library so that they can be distributed equally among students of Section A or Section B?

Assertion (A) : H.C.F. ($36 \text{ m}^2$, 18 m) = 18 m, where m is a prime number.
Reason (R) : H.C.F. of two numbers is always less than or equal to the smaller number.

If HCF of $66$ and $99$ is expressible in the form of $55m - 132$, then the value of $m$ is :

If the HCF of two positive integers a and b is $1$, then their LCM is :

Assertion (A) : For any two natural numbers a and b, the HCF of a and b is a factor of the LCM of a and b.
Reason (R) : HCF of any two natural numbers divides both the numbers.

The value of $(\text{HCF} - \text{LCM})$ for the two numbers $3$ and $5$ is :

If $p = 2^3 \times 3^2 \times 5$ and $q = 2^2 \times 3^3$, then the LCM of $p$ and $q$ is :

If $HCF(x, 20) = 2$ and $LCM(x, 20) = 60$, then value of $x$ is :

The LCM of two numbers is $3600$. Which of the following can not be their HCF ?

The HCF of $2^2.3^3$ and $3^2.2^3$ is :

The LCM of $2^2.3^3$ and $3^2.2^3$ is:

The HCF of $3^7 \cdot 7^3$ and $3^3 \cdot 7^7$ is:

The HCF of the smallest prime number and the smallest 3-digit number is $2^m 5^n$. The respective values of $m$ and $n$ are :

(i) Which of the following statements is true for HCF and LCM of two distinct natural numbers $a$ and $b$?
HCF is always greater than LCM.
(ii) HCF is a factor of LCM.
(iii) LCM is a factor of HCF.

If the HCF of the smallest odd prime number and the greatest $2$-digit number is expressed as $3^m \cdot 11^n$, then the values of $m$ and $n$ respectively are:

The LCM of the smallest $2$-digit number and the smallest $3$-digit number is expressed in the form $2^P \cdot 5^q$. The respective values of $p$ and $q$ are :

The HCF of $2^3$ and $3^2$ is:

The LCM of two consecutive natural numbers $p$ and $p + 1$ is:

The HCF of $4^3$ and $6^2$ is :

The HCF of $3^5$ and $5^3$ is :

If HCF $(850, 325)$ is $25$, then LCM $(850, 325)$ is :

HCF of two consecutive natural numbers is :

The product of the H.C.F. and L.C.M. of two numbers $50$ and $20$ is

Find the HCF and LCM of $84, 90$ and $120$ by prime factorization method.

Two numbers are in the ratio $2 : 3$ and their LCM is $180$. What is the HCF of these numbers ?

Using prime factorisation, find HCF and LCM of $96$ and $120$.

Find the HCF and LCM of $72$ and $120$.

OR
Find the LCM and HCF of $72$ and $120$

If LCM of $51$ and $85$ can be expressed in the form of $6z - 9$, then find the value of $z$.

Two numbers are in the ratio $4: 5$ and their HCF is $11$. Find the LCM of these numbers.

Find HCF and LCM of $35$ and $55$ and verify your answer.

If the HCF of $210$ and $55$ is expressed as $210 \times 5 + 55m$, then find the value of $m$.

Find the greatest number less than $10,000$ which is exactly divisible by $48, 60$ and $65$.

Two numbers are in the ratio $3 : 5$ and their LCM is $180$. Find the HCF of these two numbers.

Using prime factorisation, find the HCF of 180, 140 and 210.

Using prime factorisation, find the HCF of $144, 180$ and $192$.

Find the H.C.F. and L.C.M. of $1530$ and $2040$.

If H.C.F. of $135x^2$ and $189x^3$ is $108$, then find the value of $x$.

Find the H.C.F. and L.C.M. of $408$ and $312$.

Find by prime factorisation the LCM of the numbers $18180$ and $7575$. Also, find the HCF of the two numbers.

Find the HCF and LCM of $26, 65$ and $117$, using prime factorisation.

Let $x$ and $y$ be two distinct prime numbers and $p = x^2 y^3$, $q = xy^4$, $r = x^5 y^2$. Find the HCF and LCM of $p, q$ and $r$. Further check if HCF $(p, q, r) \times$ LCM $(p, q, r) = p \times q \times r$ or not.

The traffic lights at three different road crossings change after every $45$ seconds, $75$ seconds and $60$ seconds respectively. If they change together at $5.00$ a.m., then at what time they will change together next?

Three measuring rods are of lengths $120 \text{ cm}, 100 \text{ cm}$ and $150 \text{ cm}$. Find the least length of a fence that can be measured an exact number of times, using any of the rods. How many times each rod will be used to measure the length of the fence?

If 'n' is a natural number, then which of the following numbers end with zero?

Assertion (A): The number $5^{\text{n}}$ cannot end with the digit $0$, where $n$ is a natural number.
Reason (R): Prime factorisation of $5$ has only two factors, $1$ and $5$.

Can the number $(15)^n$, $n$ being a natural number, end with the digit $0$? Give reasons.

Assertion (A): $4^n$ ends with digit $0$ for some natural number $n$.
Reason (R) : For a number 'x' having $2$ and $5$ as its prime factors, $x^n$ always ends with digit $0$ for every natural number $n$.

Assertion (A) : Unit digit of $3^n$ cannot be an even number for any natural number $n$. Reason (R) : $2$ is not a prime factor of $3^n$ for any natural number $n$.

Which of the following cannot be the unit digit of $8^n$, where $n$ is a natural number?

For any natural number $n$, $6^n$ ends with the digit :

For any natural number $n$, $5^n$ ends with the digit:

The number $2^n$, where $n$ is a natural number, cannot end with the digit :

The number $3^n$, where $n$ is a natural number, cannot end with the digit :

If the number $a^n$, where $n$ is a natural number, always ends with digit $a$, then the possible value of '$a$' is :

Which of the following numbers will not end with 0 for any natural
number $n$ ?

Assertion (A): $4^n$ can not end with the digit zero.
Reason (R): Prime factorisation of $4^n$ is unique.

Check whether $6^n$ can end with the digit $0$ for any natural number $n$.

Prove that $4^n$ can never end with digit $0$, where $n$ is a natural number.

Show that $6^n$ can not end with digit $0$ for any natural number 'n'.

Check whether there is any natural number 'n' for which $(14)^n$ ends with the digit '0' or '5'.

Show that $14^n$ cannot end with the digit $0$ or $5$ for any natural number $n$.

Show that $45^n$ can not end with the digit $0$, $n$ being a natural number. Write the prime number '$a$' which on multiplying with $45^n$ makes the product end with the digit $0$.

Check whether $15^n \times 2^n$, n being a natural number, ends with the digit zero.

Prove that, for a natural number $n$, $6^n$ can not end with the digit $0$. Which prime number must be multiplied with $6^n$ so that the resultant ends with the digit zero?

The natural number $2$ is:

The natural number $1$ is:

$(3 \times 11 \times 13 + 3)$ is :

$(3 \times 11 \times 13 + 3)$ is :

$(3 \times 11 \times 13 + 3)$ is :

A prime number has :

The sum of first five prime numbers is

Assertion (A): $7 \times 2+3$ is a composite number.
Reason (R): A composite number has more than two factors.

$7 \times 29 \times 23 + 1$ is:

$7 \times 29 \times 23 + 1$ is:

$7 \times 29 \times 23 + 1$ is:

$7 x 11 x 13 + 5$ is

$7 \times 11 \times 13 + 5$ is

$17 \times 11 \times 13 + 11$ is

Show that the number $5\times 11\times 17+3\times 11$ is a composite number.

Explain why $7 \times 11 \times 13 + 13$ and $7\times 6\times 5\times 4\times 3\times 2\times 1+ 5$ are composite numbers.

State true or false for each of the following statements and justify in each case: (i) $2 \times 3 \times 5 \times 7 + 7$ is a composite number. (ii) $2 \times 3 \times 5 \times 7 + 1$ is a composite number.

Let $p, q$ and $r$ be three distinct prime numbers. Check whether $p \cdot q \cdot r + q$ is a composite number or not. Further, give an example for 3 distinct primes $p, q, r$ such that (i) $p \cdot q \cdot r + 1$ is a composite number. (ii) $p \cdot q \cdot r + 1$ is a prime number.

Assertion (A): The perimeter of $\triangle ABC$ is a rational number. Reason (R): The sum of the squares of two rational numbers is always rational.

If $p^2 = \frac{32}{50}$, then $p$ is a/an

A pair of irrational numbers whose product is a rational number is :

The smallest irrational number by which $\sqrt{20}$ should be multiplied so as to get a rational number, is:

$2.35$ is:

$(1+\sqrt{3})^2 -(1-\sqrt{3})^2$ is:

$(1 + \sqrt{3})^2 - (1 - \sqrt{3})^2$ is :

$(\sqrt{3}+2)^2+(\sqrt{3}-2)^2$ is a/an

For any prime number $p$, if $p$ divides $a^2$, where $a$ is any real number then $p$ also divides

$\sqrt{0.4}$ is a/an

Assertion (A): $(\sqrt{3} + \sqrt{5})$ is an irrational number. Reason (R) : Sum of the any two irrational numbers is always irrational.

The number $3 + \sqrt{2}$ is :

$\frac{\sqrt{3} - 3}{\sqrt{3}}$ is :

$(2 + \sqrt{2})^2$ is :

$(2 - 5\sqrt{3})^2$ is

$(2 - 5\sqrt{3})^2$ is

$\sqrt{2}(\sqrt{2} - 1)$ is

Assertion (A) : $(a + \sqrt{b}) \cdot (a - \sqrt{b})$ is a rational number, where $a$ and $b$ are positive integers.
Reason (R) : Product of two irrationals is always rational.

Assertion (A): $(\sqrt{2} + \sqrt{3})$ is an irrational number.
Reason (R): The sum of two irrational numbers is always an irrational number.

Assertion (A): $(\sqrt{3} + 1)^2$ is a rational number.
Reason (R): $(\sqrt{3})^2 = 3$ is a rational number.
Select the correct answer to these questions from the codes (A), (B), (C) and (D) as given below.
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
(C) Assertion (A) is true, but Reason (R) is false.
(D) Assertion (A) is false, but Reason (R) is true.

Assertion (A) : $(2 - \sqrt{5})^2$ is a rational number.
Reason (R): $(\sqrt{5})^2$ is a rational number.

Assertion (A): $(3 + 2\sqrt{2})^2$ is a rational number.
Reason (R): $\sqrt{2}$ is an irrational number.

If $\sqrt{2}$ is given as an irrational number, then prove that $(5-2\sqrt{2})$ is an irrational number.

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

Prove that $6 - \sqrt{7}$ is irrational number, given that $\sqrt{7}$ is an irrational number.

Prove that $7 - 3\sqrt{5}$ is an irrational number, given that $\sqrt{5}$ is an irrational number.

Find whether each of the following is an irrational number or a rational number.
(i) $(\sqrt{5}-3)^2$
(ii) $(5+\sqrt{3})(5-\sqrt{3})$

Prove that $5-2\sqrt{3}$ is an irrational number. It is given that $\sqrt{3}$ is an irrational number.

If $\sqrt{7}$ is an irrational number, then prove that $2\sqrt{7}$ is also an irrational number.

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

Prove that $2 - 5\sqrt{3}$ is an irrational number given that $\sqrt{3}$ is irrational.

Prove that $4 - 2\sqrt{5}$ is an irrational number given that $\sqrt{5}$ is irrational.

Prove that $14 - 2\sqrt{3}$ is an irrational number, given that $\sqrt{3}$ is irrational.

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

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

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

Prove that $\frac{2-\sqrt{3}}{5}$ is an irrational number, given that $\sqrt{3}$ is an irrational number.

Prove that $(\sqrt{2} + \sqrt{3})^2$ is an irrational number, given that $\sqrt{6}$ is an irrational number.

Prove that $\frac{1}{\sqrt{5}}$ is an irrational number.

This section has $6$ Short Answer (SA) type questions carrying $3$ marks each.
Prove that $\sqrt{5}$ is an irrational number.

Prove that $(5\sqrt{3} + \frac{2}{3})$ is an irrational number given that $\sqrt{3}$ is an irrational number.

Prove that $\left(4\sqrt{2} + \frac{5}{3}\right)$ is an irrational number given that $\sqrt{2}$ is an irrational number.

Prove that $\frac{2 + 3\sqrt{5}}{7}$ is an irrational number, given that $\sqrt{5}$ is an irrational number.

State the "Fundamental Theorem of Arithmetic" and use it to find LCM of $36$ and $54$.

Find which among the following numbers $a, b$ and $c$ is/are composite numbers.
$a = 7 \times 11 \times 13 + 13$
$b = 6 \times 5 \times 4 + 4$
$c = 7 \times 13 + 6$

Given that $\sqrt{5}$ is an irrational number, prove that $2 + 3\sqrt{5}$ is an irrational number.

Prove that $\frac{\sqrt{3}+5}{2}$ is an irrational number, where it is given that $\sqrt{3}$ is irrational.

This section comprises $6$ Short Answer (SA) type questions of $3$ marks each.
Given that $\sqrt{5}$ is an irrational number, prove that $3+2\sqrt{5}$ is also an irrational number.

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

Given that $\sqrt{3}$ is an irrational number, prove that $2-5\sqrt{3}$ is also an irrational number.

Neha claimed that there does not exist any irrational number between 1 and 2. Raunak claimed that $\sqrt{2}$ lies between 1 and 2 and $\sqrt{2}$ is an irrational number. Who do you think is correct? Justify by proving either $\sqrt{2}$ as an irrational number or otherwise.

The greatest number which divides both $83$ and $138$, leaving remainders $5$ and $8$ respectively, is :

The LCM of smallest $2$-digit number and smallest composite number is

The greatest number which divides $281$ and $1249$, leaving remainder $5$ and $7$ respectively, is :

The LCM of the smallest prime number and the smallest odd composite number is:

If $(-1)^n + (-1)^8 = 0$, then $n$ is:

The greatest number which divides $70$ and $125$, leaving remainders $5$ and $8$ respectively, is:

The least number which is a perfect square and is divisible by each of $16$, $20$ and $50$, is :

Find the greatest number which divides 85 and 72 leaving remainders 1 and 2 respectively.

Find the least number which when divided by 12, 16 and 24 leaves remainder 7 in each case

Find the greatest $3$-digit number which is divisible by $18, 24$ and $36$.

Find the smallest $4$-digit number exactly divisible by $15$, $24$ and $36$.

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?

In a school, there are two sections of class X. There are $40$ students in the first section and $48$ students in the second section. Determine the minimum number of books required for their class library so that they can be distributed equally among students of both sections.

Find the smallest number that is divisible by each of $8, 9$ and $10$.

Find the smallest number which is divisible by both $644$ and $462$.

Find the length of the plank that can be used to measure the lengths $4$ m $20$ cm and $5$ m $4$ cm exactly, in the least time.

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

Three bells ring at intervals of $6$, $12$ and $18$ minutes. If all the three bells rang at $6$ a.m., when will they ring together again?

The traffic lights at three different road crossings change after every $48$ seconds, $72$ seconds and $108$ seconds respectively. If they change simultaneously at $7$ a.m., at what time will they change together next?

A school has invited $42$ Mathematics teachers, $56$ Physics teachers and $70$ Chemistry teachers to attend a Science workshop. Find the minimum number of tables required, if the same number of teachers are to sit at a table and each table is occupied by teachers of the same subject.

In a teachers' workshop, the number of teachers teaching French, Hindi and English are $48, 80$ and $144$ respectively. Find the minimum number of rooms required if in each room the same number of teachers are seated and all of them are of the same subject.

Ranjita, Neha and Salma start weaving sweaters at the same time for the children of an orphan home. They need $15, 18$ and $20$ days, respectively, to complete a sweater. After how many days will all of them start making a new sweater again? By that time how many sweaters will have been competed by them?

Three sets of Physics, Chemistry and Mathematics books have to be stacked in such a way that all the books are stored subject-wise and the height of each stack is the same. The number of Physics books is $144$, the number of Chemistry books is $180$ and the number of Mathematics books is $192$. Assuming that the books are of same thickness, determine the number of stacks of Physics, Chemistry and Mathematics books.

A trader has three different types of oils of volume $870$ l, $812$ l and $638$ l. Find the least number of containers of equal size required to store all the oil without getting mixed.

Find the greatest number which divides $764$ and $1198$, leaving remainders $8$ and $10$ respectively.

The dimensions of a window are $156$ cm $\times 216$ cm. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

Find the smallest number which when increased by $20$, is exactly divisible by $72, 90$ and $150$.

Three friends plan to go for a morning walk. They step off together and their steps measures $48$ cm, $52$ cm and $56$ cm respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps ten times?

February $14$ is celebrated as International Book Giving Day and many countries in the world celebrate this day. Some people in India also started celebrating this day and donated the following number of books of various subjects to a public library :
History = $96$, Science = $240$, Mathematics = $336$.
These books have to be arranged in minimum number of stacks such that each stack contains books of only one subject and the number of books on each stack is the same.
Based on the above information, answer the following questions :
(i) How many books are arranged in each stack?
(ii) How many stacks are used to arrange all the Mathematics books?
(iii) (a) Determine the total number of stacks that will be used for arranging all the books.
OR
(iii) (b) If the thickness of each book of History, Science and Mathematics is $1.8$ cm, $2.2$ cm and $2.5$ cm respectively, then find the height of each stack of History, Science and Mathematics books.

Teaching Mathematics through activities is a powerful approach that enhances students' understanding and engagement. Keeping this in mind, Ms. Mukta planned a prime number game for class $5$ students. She announces the number $2$ in her class and asked the first student to multiply it by a prime number and then pass it to second student. Second student also multiplied it by a prime number and passed it to third student. In this way by multiplying to a prime number, the last student got $173250$.
Now, Mukta asked some questions as given below to the students :
(i) What is the least prime number used by students?
(ii) (a) How many students are in the class ?
OR
(b) What is the highest prime number used by students?
(iii) Which prime number has been used maximum times ?

This section has $3$ case study based questions carrying $4$ marks each.
Case Study – 1
The science department of a college is conducting an international seminar in which the number of participants in Physics, Chemistry and Biology are $65, 91$ and $117$ respectively. The coordinator has made the arrangement such that in each room, the same number of participants are to be seated with all of them being in the same subject.
Based on the information given above, answer the following questions :
(i) Find the HCF of $65, 91$ and $117$.
(ii) Find the LCM of $65, 91$ and $117$.
(iii) (a) Find the minimum number of rooms required based on the above conditions.
OR
(iii) (b) Find the minimum number of participants to be accommodated in each of the rooms.

Assertion (A): A fair die is thrown once. The probability of getting a prime number is $\frac{1}{2}$.
Reason (R): A natural number is a prime number if it has only two factors.

Statement A (Assertion) : If $5 + \sqrt{7}$ is a root of a quadratic equation with rational co-efficients, then its other root is $5 - \sqrt{7}$.
Statement R (Reason) : Surd roots of a quadratic equation with rational co-efficients occur in conjugate pairs.

A natural number, when increased by $12$, equals $160$ times its reciprocal. Find the number.