1) Determine whether the following functions are well-defined:
1a) defined by
1b) defined by
2) Determine whether the function defined by mapping a real number r to the first digit to the right of the decimal point in a decimal expansion of r is well-defined.
3) Apply the Euclidean algorithm to obtain GCD of and express it as a linear combination of 57970 and 10353.
4) For each of the following pairs of integers a and b, determine their greatest common divisor, their least common multiple, and write their greatest common divisor in the form for some integers x and y.
(a) a=20, b=13
(b) a=69, b=372
(c) a=792, b=275
(d) a=11391, b=5673
(e) a=1761, b=1567
(f) a=507885, b=60808
5) Prove that if the integer k divides the integers a and b then k divides for every pair of integers s and t.
6) Prove that if n is composite then there are integers a and b such that a divides ab but n does not divide either a or b.
7) Let a, b and N be fixed integers with a and b non-zero and let be the greatest common divisor of a and b. Suppose and are particular solutions to . Prove for any integer r that integers and are also solutions to (this is in fact the general solution).
8) Determine the value for each integer where denotes the Euler- function.
9) Prove the Well-Ordering Property of integers by induction and prove the minimal element is unique.
10) If p is a prime prove that there do not exist non-zero integers a and b such that (that is, is not a rational number).
11) Let p be a prime and . Find a formula for the largest power of p which divides (it involves the greatest integer function).
12) Prove for any given positive integer N there exist only finitely many integers n with where denotes Euler’s -function.
13) Prove that if d divides n then $latex \phi(d)$ divides where denotes Euler’s -function.
Hope this gives you some math meal to churn for the Pre RMO or PRMO or even the ensuing RMO of Homi Bhabha Science Foundation.