Show that the square of any positive integer is either of the form \(4q\) or \(4q + 1\) for some integer \(q\).
Squares are of the form \(4q\) or \(4q+1\).
Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
Cubes are of the form \(4m\), \(4m+1\) or \(4m+3\).
Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.
Impossible to be \(5q+2\) or \(5q+3\).
Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.
Impossible to be \(6m + 2\) or \(6m + 5\).
Show that the square of any odd integer is of the form 4q + 1, for some integer q.
\(4q + 1\)
If n is an odd integer, then show that \(n^2 - 1\) is divisible by 8.
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 \(4\).
Even, not divisible by 4.
Use Euclid’s division algorithm to find the HCF of 441, 567, 693.
63
Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.
625
Prove that \(\sqrt{3}+\sqrt{5}\) is irrational.
Irrational.
Show that 12n cannot end with the digit 0 or 5 for any natural number n.
Cannot end with 0 or 5.
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?
2520 cm (i.e., 25.2 m)
Write the denominator of the rational number \(\dfrac{257}{5000}\) in the form 2m × 5n, where m, n are non-negative integers. Hence, write its decimal expansion, without actual division.
Denominator: \(2^3 \cdot 5^4\). Decimal: 0.0514
Prove that \(\sqrt{p}+\sqrt{q}\) is irrational, where \(p, q\) are primes.
Irrational.