Consider the eight-digit bank identification number , which is followed by a ninth check digit chosen to satisfy the congruence
(a) Obtain the check digits that should be appended to the two numbers 55382006 and 81372439.
(b) The bank identification number has an illegitimate fourth digit. Determine the value of the obscured digit.
(a) Find an integer having the remainders 1,2,5,5 when divided by 2, 3, 6, 12 respectively (Yih-hing, died 717)
(b) Find an integer having the remainders 2,3,4,5 when divided by 3,4,5,6 respectively (Bhaskara, born 1114)
(c) Find an integer having remainders 3,11,15 when divided by 10, 13, 17, respectively (Regiomontanus, 1436-1476)
Let denote the nth triangular number. For which values of n does divide
Hint: Because , it suffices to determine those n satisfying
Find the solutions of the system of congruences:
Obtain the two incongruent solutions modulo 210 of the system
Use Fermat’s Little Theorem to verify that 17 divides
(a) If , show that . Hint: From Fermat’s Little Theorem, and
(b) If , show that divides
(c) If , show that
Show that and . Do there exist infinitely many composite numbers n with the property that and ?
Prove that any integer of the form is an absolute pseudoprime if all three factors are prime; hence, is an absolute pseudoprime.
Prove that the quadratic congruence , where p is an odd prime, has a solution if and only if .
Note: By quadratic congruence is meant a congruence of the form with . This is the content of the above proof.