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Problem Session 1
1. Calculate (243, 40) and find x and y such that 243x + 40y = 1.
2. Prove: If a|b and c|d, then ac|bd.
3. Let (a , b) = d and suppose that c|a and c|b. Prove that c|d. (Hint: think about the
purpose of the Extended Euclidean Algorithm.)
0 a
0
4. Consider the expressions ,
, and
for a  0 in light of the definition of divides.
a 0
0
a. Write the corresponding divisibility statement for each expression. (For
example, a0 corresponds to the divisibility statement a|0.) Using the definition
of divides, determine which of these divisibility statements are true.
b. Only one expression determines a unique numerical value. Using your results
from part (a), explain why the other two don’t.
c. Now consider the fraction 72 . Notice that its corresponding divisibility
statement is false. Does this mean that 72 is undefined? Compare this with the
expression a0 .
5. Fun with gcd’s.
a. Prove that for every positive integer a, (a, a + 2) is always either 1 or 2.
b. For what positive integers a does (a, a + 2) = 2? (Make a conjecture and try to
prove it.)
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
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Problem Session 2
1. Use the prime decompositions of 111111 and 34560, find (111111, 34560).
2. Find the smallest integer divisible by 2 and 3 which is simultaneously a square and a
fifth power.
3. If d|ab, does if follow that d|a or d|b?
4. Prove that n(n + 1) is never a square for n >0. (Hint: n 2 and (n  1) 2 are consecutive
squares.)
5. Is it possible for a prime p to divide both n and n + 1?
6. Prime free gaps
a. True or False:
i. 7 | (9! + 7)
ii. 7 | (33! + 7)
b. Find 6 consecutive natural numbers, none of which is prime. (Hint: consider
the number 7! + 2.)
c. Find 7854 consecutive natural numbers, none of which is prime.
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
2
Problem Session 3
1. Consider the equation 3x – 4y = 5.
a. Find one (integer) solution pair.
b. Use the slope of the line to plot 6 additional (integer) solution pairs. Label the
solutions on your graph.
c. Write the form of the integer solutions.
d. Verify that the solutions you found in part c) satisfy the equation.
2. Person A and Person B are studying the practicalities of a monetary system which utilizes
seven and nine dollar bills (no other bills are minted). Assume that each person has both
$7 and $9 bills.
a. Explain how Persons A and B should exchange bills so that Player A is paid
$3, $5, $10
b. Find a Diophantine equation whose solutions represent all possible ways that
Persons A and B can exchange bills so that a player is paid $3.
c. Find all integer solution pairs of this equation.
3. In a corral, there are cowboys and an odd number of horses. There are 80 legs in all.
a. Write a Diophantine equation to describe this situation.
b. Use the Diophantine equation to find the possible numbers of cowboys and
horses that could be in the corral.
4. Find all integer solutions of
a. 10x + 4y = 6
b. 10x + 4y = 5
c. 6x + 15y = 3
d. 12x + 4y = 16
e. 15x + 16y = 17
f. 885x + 330y = 15
g. 1891x + 337y = 1
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
3
Problem Session 4
1. Find the least residue of 1492 (mod 4), (mod 10), and (mod 101). Find the least residue
of –5 (mod 13) and (mod 65).
2. Compute 1753247  235711131719 (mod 5)
3. Prove or disprove that if a  b (mod m), then a 2  b 2 (mod m).
4. In the multiplication problem 31415  92653 = 2910w93995, the w is missing. Find w
without doing the multiplication.
5. Show that no square has as its last digit 2, 3, 7, or 8.
6. Show that the difference of two consecutive cubes is never divisible by 3.
7. (a) Show that
d k 10 k  d k 110 k 1    d110  d 0  d 0  d1  d 2  d 3    (1) k d k (mod 11)
(b) State a divisibility test for 11 that your middle school students would understand.
(c) Use your test to show that 7,454,623 is divisible by 11.
8. The Canadian province of Quebec assigns a check digit a12 to an 11-digit driver’s license
number a1a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 so that
12a1  11a2  10a3  9a4  8a5  7a6  6a7  5a8  4a9  3a10  2a11  a12
is divisible by 10. Describe all single-digit errors that are undetected by this scheme.
9. Divisibility tests:
a. Prove: If the last 2 digits of a natural number is divisible by 4, then the number
is divisible by 4.
b. Prove: If the last 3 digits of a natural number is divisible by 8, then the number
is divisible by 8.
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
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Problem Session 5
1. Solve each of the following equations without using the Extended Euclidean
Algorithm.
a. 2x  1 (mod 17)
b. 3x  1 (mod 17)
c. 3x  6 (mod 18)
2. Solve the following equations by using the Extended Euclidean Algorithm, and list the
smallest three positive solutions.
a. 40x  1 (mod 243) (Refer to Problem Session #1)
b. 40 x  47 (mod 243)
c. 53x  4 (mod 169)
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
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Problem Session 6
1. Compute each of the following:
a. 117 (mod 21)
b. 3 42 (mod 5)
c. 7 53 (mod 15)
d. 6 74 (mod 13)
2. Show that the converse of Fermat’s Theorem is false. (Broad hint: consider 2 340 (mod
341).)
3. Suppose we have a deck of 52 playing cards and we cut it into two equal piles, the top
half and the bottom half. If we then interlace the cards, with the bottom card of the top
half going on the bottom of the shuffled pile, the bottom card of the bottom half going
on top of this card, etc., we call this process an in-shuffle (because the original top and
bottom cards of the deck are inside the deck after the shuffle).
a. If a card starts out x spots from the bottom of the original deck, where does it
end up after one perfect in-shuffle?
b. If a card starts at position x (counting from the bottom of the deck), wherer
does it end up after 2 perfect in-shuffles? 3 perfect in-shuffles? k perfect inshuffles?
c. Is there a positive number of shuffles that one can do which will restore the
entire deck to its original position?
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
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Problem Session 7
1. Calculate  (35) and  (83) .
2. Calculate a 8 (mod 15) for a = 1, 2, …, 14. (Hint: is there a theorem that can save you
some time in computing this for some of the a’s?)
3. Let  (n) denote the number of divisors of n. For example,  (8)  4 since 1, 2, 4 and 8
are divisors of 8.
a. Compute  (36)
b.
c.
d.
e.
f.
g.
Compute  (5 3 7 4 )
Compute  (5 a 7 b ) , where a and b are positive integers.
Compute  (2 a 3b 5c ) , where a, b, and c are positive integers.
Classify all n for which  (n) = 2.
Classify all n for which  (n) = 6.
Classify all n for which  (n) = 7.
4. Let s(n) denote the sum of the divisors of n that are less than n. For example,
s(8) = 1 + 2 + 4 = 7.
a. Compute s(10) and s(13)
b. If p is prime, what is s( p) , s( p 2 ) , s( p 3 ) and s( p n ) ?
c. Perfect numbers are numbers n satisfying s(n) = n; in other words, the sum of
the divisors less than the number is the number itself. Find two perfect
numbers.
Copyright 2007. Number Theory and Cryptology for Middle Level Teachers. Developed by the Math in the Middle Institute
Partnership, University of Nebraska, Lincoln.
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