NCERT Exemplar solutions for Mathematics [English] Class 10 chapter 1 - Real Numbers [Latest edition]

For some integer m, every even integer is of the form ______.

For some integer q, every odd integer is of the form ______.

n² – 1 is divisible by 8, if n is ______.

If the HCF of 65 and 117 is expressible in the form 65m – 117, then the value of m is ______.

The largest number which divides 70 and 125, leaving remainders 5 and 8, respectively, is ______.

If two positive integers a and b are written as a = x³ y² and b = xy³; x, y are prime numbers, then HCF (a, b) is ______.

If two positive integers p and q can be expressed as p = ab² and q = a³b; a, b being prime numbers, then LCM (p, q) is ______.

The product of a non-zero rational and an irrational number is ______.

The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is ______.

The decimal expansion of the rational number `14587/1250` will terminate after ______.

Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify your answer.

“The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.

“The product of three consecutive positive integers is divisible by 6”. Is this statement true or false”? Justify your answer.

Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer.

A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.

The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525, 3000)? Justify your answer.

Explain why 3 × 5 × 7 + 7 is a composite number.

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

Without actually performing the long division, find if `987/10500` will have terminating or non-terminating (repeating) decimal expansion. Give reasons for your answer.

A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the form `p/q`? Given reasons.

Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.

Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.

Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.

Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.

Show that the square of any odd integer is of the form 4q + 1, for some integer q.

If n is an odd integer, then show that n² – 1 is divisible by 8.

Prove that if x and y are both odd positive integers, then x² + y² is even but not divisible by 4.

Use Euclid’s division algorithm to find the HCF of 441, 567, 693.

Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.

Prove that `sqrt(3) + sqrt(5)` is irrational.

Show that 12ⁿ cannot end with the digit 0 or 5 for any natural number n.

On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

Write the denominator of the rational number `257/5000` in the form 2^m × 5ⁿ, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.

Prove that `sqrt(p) + sqrt(q)` is irrational, where p, q are primes.

Show that the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.

Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.

Prove that one of any three consecutive positive integers must be divisible by 3.

For any positive integer n, prove that n³ – n is divisible by 6.

Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.

[Hint: Any positive integer can be written in the form 5q, 5q + 1, 5q + 2, 5q + 3, 5q + 4].