Mathematics exemplar problems class 10 Solutions for Chapter 1 real numbers

n^2-1 is divisible by 8, if n is

For some integer m, every even integer is of the form:

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

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

If two positive integers p and q can expressed as p = ab^ 2 \ and\ q = a^3b where a and b being prime numbers, then LCM (p, q) is

If two positive integers a and b are written as a = x^3y ^2 \ and \ b = cy^3 ; x, y are prime numbers then HCF (a, b) 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 will terminate 14587/1250 after

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

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

State 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?

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

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

A positive integer is of the form 3q + 1, q being a natural number. In which of the following can you write its square other than 3m + 1?

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 be concluded about the prime factors of q, when this number is expressed in the form p/q? Give reasons.

State whether 3 × 5 × 7 + 7 is a composite number.

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

State whether the two numbers can have 18 as their HCF and 380 as their LCM? Give reasons.

If x and y are both odd positive integers, then x² + y²

is even but not divisible by 4. State whether true or false.

n is an odd integer, then n^2 – 1 is divisible by 8. State whether the statment is true or false.

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

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

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

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

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

Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 1562 8 leaving remainders, 1, 2, and 3 respectively.

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

12^n (n is any natural number), cannot end with which of the following digits?

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

Check if \sqrt p+ \sqrt q is irrational or rational, where p and q are primes.

State whether if \sqrt{3}+\sqrt{5} is irrational or rational. Check using method of contradiction.

Write the denominator of rational number \frac{257}{5000} in the form of 2^m\times5^n, where m, n are non-negative integers. Hence, ffind what will be its decimal expansion, without actual division

One and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer. State whether true or false.

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. State whether the statment is true or false. Justify.

One of any three consecutive positive integers must be divisible by 3. State whether true or false.

For any positive integer n, n^3 – n is divisible by 6. State whether true or false.

One and only one out of n, (n + 4), (n + 8), (n + 12), (n + 16) is divisible by 5, where n is any positive integer. State whether the statment is true or false.