Show that the square of an odd positive integer is of the form 8q + 1, for some integer q.

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

If the H.C.F. of 210 and 55 is expressible in the form 210 x 5 + 55y. Find y.

lf d is the H.C.F. of 56 and 72, find x and y satisfying d = 56x + 72y

In an Interview, the number of participants in Mathematics, Physics and Chemistry are 60, 84 and 108 respectively. Find the minimum number of rooms required if in each room the same number of participants to be seated and alt of them being in the same subject

Find the greatest number which can divide 1356, 1868 and, 2764 leaving the same remainder 12 in each case

Find the smallest number of five digits exactly divisible by 16,24,36 and 54.

Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10 and 12 seconds respectively In 30 minutes, how many times do they toll together?

\(\text { Solve } 9 x \equiv 5(\bmod 13)\)

Find the next five terms of the sequence given by \(a_{1}=4 ; a_{n}=4 a_{n-1}+3, n>1\)

A sequence is defined by \(a_{n}=n^{3}-6 n^{2}+11 n-6\) show that the First three terms of the sequence are zero and all other terms are positive

The 10^th term an A.P. is 52 and 16^th term is 82. Find the 32^nd term and the general term.

The sum of 5^th and 9^th terms of an A.P. is 72 and the sum of 7^th and 12^th term is 97. Find the A.P.

If \(\frac{1}{b+c}, \frac{1}{c+a}, \frac{1}{a+b} \text { are in A.P. then prove that }\mathrm{a}^{2}, \mathrm{~b}^{2}, \mathrm{c}^{2} \text { are in A.P }\)

Find the 6^th term of the A.P. \(\frac{2 m+1}{m}, \frac{2 m-1}{m}, \frac{2 m-3}{m}, \ldots\)