Ayush

Given an integer N , express it as a^b where a is smallest possible positive integer and b is the number of divisors of b.Hence given N , find a and b.

For eg: if N=25 , a=5 , b=2{1,2}

Note:It may be possible that N cannot be expressed as a^b for any value of a and b

The points can be in or on the circumference of the circle.

This is a programming question ,if that helps.

There are n numbers. Find the maximal set of pairwise NON coprime numbers.

Ex:- 2,3,4,9,15,16,45,81,27

One such set - 2,4,16 but not maximal

another set - 3,15,27,45,81 it is maximal

I want to find the no, of elements in this maximal set , i.e 5.

Example if n=3 , numbers are 4 5 6 , least lcm = 12 (lcm of 4, 6)

I have to do this for a large n and large numbers.

I know I can take every possible pair and find lcm for each and then see the min lcm, but i need a more faster and efficient method.

In a village there are n houses , some have electricity some dont. Those having electricity are labeled as 1 and others 0.

Houses are arranged in a line from left to right (1 to n).

let x1 x2 x3..... be coordinates of the houses.(x1<x2<x3.......<xn)

The houses having electricity are to be connected with those which don't have.Find the minimum length of wire required.

Example:No. of houses n=3

Label 100

coordinates 1,5,6

Minimum length req= 6-1=5 (connect first and third house. The wire will go through 2nd house thus all have electricity)

2)label 10100

coordinates 1,4,6,9,88

Minimum length = 84[(6-4)+(88-6)]

Take the houses to be arranged as if on x axis and the coordinates need not start from 1 but 1 is the lower limit, i.e

1<=x1<x2<x3.......<xn

Let two circles cut each other at origin O at an angle of pi/3 and the common tangent opposite to O touches the circles at A and B. If combined equation of OA and OB is ax²+2hxy + ay²=0 , then

(A) h²=4a²

(B)2h²=a²

(C)3h²=4a²

(D)4h²=a²

The mixture of two immiscible liquids (water and nitrobenzene) boils at 372 K and the vapour pressure at this temperature are 97.7 kPa and 3.6 pka

Ans = 20.11%