Advanced High-School Mathematics

74 CHAPTER 2 Discrete Mathematics

m ≡ 7(mod 10)

m ≡ 17(mod 26).

5. Find the least positive integer solution of the congruences

m ≡ 7(mod 10)

m ≡ 5(mod 26)

m ≡ 1(mod 12).

6. Solve the problem of the woman and the eggs, given at the begin-
   ning of this section.


7. IfAandBare sets, one defines theCartesian productofAand
   Bby setting

A×B = {(a,b)|a∈Aand b∈B}.

Now suppose that the positive integers m and n are relatively

prime, and define the function

f:Zmn→Zm×Zn byf(xmn) = (xm,xn)∈Zm×Zn.

Using the Chinese remainder theorem, show that the functionfis

one-to-one and onto.

8.^8 The integerN is written as

N = 102030x 05060 y

in decimal (base 10) notation, wherex andy are missing digits.

Find the values of x and y so that N has the largest possible

value and is also divisible by both 9 and 4. (Hint: note that

N ≡−1 +x+y(mod 9) andN≡y(mod 4).)

(^8) This is problem #5 on the January 10, 2008 ASMA (American Scholastic Mathematics Associ-
ation) senior division contest.