Mathematics Hothouse

Question Paper Set A:

11) For natural numbers x and y, let denote the greatest common divisor of x and y. How many pairs of natural numbers x and y with satisfy the equation ?

Solution:

Here, if the condition, were not there, the answer would be infinitely many. But, so, just substitute some small numbers and check what is happening. In fact, in any number theory problem, first we should play with small numbers in our head.

Upon substitution, you will find that only satisfy the equation. Hence, the answer is 3.

12) Let ABCD be a convex quadrilateral with

. Let the incircles of triangles ABD and BCD touch  BD at P and Q respectively, with P lying in between B and Q. If

and then what is the sum of  the radii of the incircles of triangles ABD and BDC?

Solution: Please download the jpg fig attached.

The main properties to  be used are Pythagoras’ theorem, that the angle bisectors of the vertices of a triangle meet at its incenter, the formula of a triangle.

Let and be the incenter of and

respectively. Then, as shown in  the attached figure:

Since, is a right angled isosceles triangle, , , so

. Similarly, find the other radius and sum up the two.

Question 18.

Let f be a one-to-one function from the set of natural numbers to itself such that for all natural numbers m and n. What is the least possible value of ?

Probable Solution. 

Use the Euler function. But, the answer this gives is different from the answer key.

Please send your comments, suggestions to this.

More later,

Nalin Pithwa