The International Standard Book Number (ISBN) used in many libraries consists of nine digits followed by a tenth check digit (somewhat like Hamming codes), which satisfies
Determine whether each of the ISBN’s below is correct.
(a) 0-07-232569-0 (USA)
(b) 91-7643-497-5 (Sweden)
(c) 1-56947-303-10 (UK)
When printing the ISBN , two unequal digits were transposed. Show that the check digits detected this error.
Remark: Such codes are called error correcting codes and are fundamental to wireless communications including cell phone technologies.
Question: Prove that any integer can be written as the sum of the cubes of five integers, not necessarily.
We use the identity for , which is an integer for all n. We obtain
Hence, n is equal to the sum
1) Find all prime numbers that divide 50!
2) If p and are both prime numbers, prove that is also prime.
3) (a) If p is a prime, and , prove that in the AP a, , , , , every pth term is divisible by p.
3) (b) From part a, conclude that if b is an odd integer, then every other term in the indicated progression is even.
4) Let denote the nth prime. For , show that .
Hint: Use induction and Bertrand's conjecture.
5) Prove that for every , there exists a prime p with .
Here is a cute example of the power of theory of congruences. Monster numbers can be tamed !!
Find the last two digits of .
A famous mathematician, George Polya said that a good problem solving technique is to solve an analagous less difficult problem.
So, for example, if the problem posed was “find the last two digits of 2479”. How do we go about it? Find the remainder upon division by 100. Now, how does it relate to congruences ? Modulo 100 numbers !
So, the problem reduces to — find out .
Now, what is the stumbling block…the exponent makes the whole problem very ugly. But,
, which means , that is, ,
also, use the fact
So, now we need to compute