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

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

n^{2} -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 = x^{3} y^{2} and b = xy^{3}, where 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 = a3 b; where a, b being prime numbers, them LCM (p, q) is equal to

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

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

The decimal expansion of the rational number 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) = 75?

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

Thinking Process

A number which has more than two factors is known as 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 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 formGive 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 integer is of the form 4m + 1, for some integer m.

If n is an odd integer, then show that n^{2} – 1 is divisible by 8.

Prove that, if x and y are both odd positive integers, then x^{2} + y^{2} is even but not divisible by 4.

Use Euclid’s division algorithm to find the HCF of 441, 567 and 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 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 rational number in the form 2^{m} × 5^{n}, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.

Prove that is irrational, where p and 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 n^{3}-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.