A FRIENDLY
INTRODUCTION TO
NUMBER THEORY (4TH
EDITION, 2014) –
SOLUTIONS MANUAL –
SILVERMAN
Exam: A Friendly Introduction to Number Theory (4th
Edition)
Chapter 1-4: Pythagorean Triples and Fermat's Last Theorem
1. A Pythagorean triple consists of positive integers satisfying which equation?
A. a+b=ca+b=c
B. a2+b2=c2a2+b2=c2
C. a2−b2=c2a2−b2=c2
D. a3+b3=c3a3+b3=c3
Correct Answer: B
Rationale: A Pythagorean triple consists of positive integers satisfying
a2+b2=c2a2+b2=c2 .
,2. What condition makes a Pythagorean triple "primitive"?
A. The hypotenuse is odd
B. gcd(a,b,c)=1gcd(a,b,c)=1
C. All sides are even
D. a=ba=b
Correct Answer: B
Rationale: A primitive Pythagorean triple has gcd(a,b,c)=1gcd(a,b,c)=1 .
3. The parametric formula for primitive Pythagorean triples is:
A. a=m+n,b=m−n,c=mna=m+n,b=m−n,c=mn
B. a=m2−n2,b=2mn,c=m2+n2a=m2−n2,b=2mn,c=m2+n2
C. a=m2+n2,b=2mn,c=m2−n2a=m2+n2,b=2mn,c=m2−n2
D. a=mn,b=m+n,c=m2+n2a=mn,b=m+n,c=m2+n2
Correct Answer: B
Rationale: The standard formula is
a=m2−n2,b=2mn,c=m2+n2a=m2−n2,b=2mn,c=m2+n2 with m>n>0m>n>0,
gcd(m,n)=1gcd(m,n)=1, and m,nm,n having opposite parity .
4. In a primitive Pythagorean triple, exactly one leg is even. Which leg is it?
A. a=m2−n2a=m2−n2
B. b=2mnb=2mn
C. c=m2+n2c=m2+n2
D. Both legs are odd
Correct Answer: B
,Rationale: In a primitive Pythagorean triple, exactly one leg is even, and it is the leg
given by b=2mnb=2mn .
5. If m=3m=3 and n=2n=2, the Pythagorean triple is:
A. (3, 4, 5)
B. (5, 12, 13)
C. (8, 15, 17)
D. (7, 24, 25)
Correct Answer: B
Rationale: a=32−22=9−4=5a=32−22=9−4=5; b=2(3)(2)=12b=2(3)(2)=12;
c=32+22=9+4=13c=32+22=9+4=13 .
6. What is Fermat's Last Theorem?
A. There are infinitely many prime numbers
B. xn+yn=znxn+yn=zn has no positive integer solutions for n>2n>2
C. Every even number is the sum of two primes
D. 22 is irrational
Correct Answer: B
Rationale: Fermat's Last Theorem states that xn+yn=znxn+yn=zn has no positive
integer solutions for n>2n>2 .
7. Which of the following is a primitive Pythagorean triple?
A. (6, 8, 10)
B. (9, 12, 15)
C. (8, 15, 17)
D. (10, 24, 26)
, Correct Answer: C
Rationale: (8, 15, 17) has gcd(8,15,17)=1gcd(8,15,17)=1. The others are multiples
of smaller triples .
8. For hypotenuse c=65c=65, two primitive Pythagorean triples exist. They are:
A. (16, 63, 65) and (33, 56, 65)
B. (25, 60, 65) and (39, 52, 65)
C. (20, 63, 65) and (33, 56, 65)
D. (16, 63, 65) and (25, 60, 65)
Correct Answer: A
Rationale: (16, 63, 65) comes from m=8,n=1m=8,n=1. (33, 56, 65) comes from
m=7,n=4m=7,n=4. (25, 60, 65) is not primitive .
9. In a primitive Pythagorean triple, c=m2+n2c=m2+n2 is always:
A. Even
B. Odd
C. Divisible by 3
D. A perfect square
Correct Answer: B
Rationale: If mm and nn have opposite parity, m2m2 and n2n2 are one even, one odd,
so their sum is odd .
10. The formula a=m2−n2,b=2mn,c=m2+n2a=m2−n2,b=2mn,c=m2+n2
generates all primitive Pythagorean triples when:
INTRODUCTION TO
NUMBER THEORY (4TH
EDITION, 2014) –
SOLUTIONS MANUAL –
SILVERMAN
Exam: A Friendly Introduction to Number Theory (4th
Edition)
Chapter 1-4: Pythagorean Triples and Fermat's Last Theorem
1. A Pythagorean triple consists of positive integers satisfying which equation?
A. a+b=ca+b=c
B. a2+b2=c2a2+b2=c2
C. a2−b2=c2a2−b2=c2
D. a3+b3=c3a3+b3=c3
Correct Answer: B
Rationale: A Pythagorean triple consists of positive integers satisfying
a2+b2=c2a2+b2=c2 .
,2. What condition makes a Pythagorean triple "primitive"?
A. The hypotenuse is odd
B. gcd(a,b,c)=1gcd(a,b,c)=1
C. All sides are even
D. a=ba=b
Correct Answer: B
Rationale: A primitive Pythagorean triple has gcd(a,b,c)=1gcd(a,b,c)=1 .
3. The parametric formula for primitive Pythagorean triples is:
A. a=m+n,b=m−n,c=mna=m+n,b=m−n,c=mn
B. a=m2−n2,b=2mn,c=m2+n2a=m2−n2,b=2mn,c=m2+n2
C. a=m2+n2,b=2mn,c=m2−n2a=m2+n2,b=2mn,c=m2−n2
D. a=mn,b=m+n,c=m2+n2a=mn,b=m+n,c=m2+n2
Correct Answer: B
Rationale: The standard formula is
a=m2−n2,b=2mn,c=m2+n2a=m2−n2,b=2mn,c=m2+n2 with m>n>0m>n>0,
gcd(m,n)=1gcd(m,n)=1, and m,nm,n having opposite parity .
4. In a primitive Pythagorean triple, exactly one leg is even. Which leg is it?
A. a=m2−n2a=m2−n2
B. b=2mnb=2mn
C. c=m2+n2c=m2+n2
D. Both legs are odd
Correct Answer: B
,Rationale: In a primitive Pythagorean triple, exactly one leg is even, and it is the leg
given by b=2mnb=2mn .
5. If m=3m=3 and n=2n=2, the Pythagorean triple is:
A. (3, 4, 5)
B. (5, 12, 13)
C. (8, 15, 17)
D. (7, 24, 25)
Correct Answer: B
Rationale: a=32−22=9−4=5a=32−22=9−4=5; b=2(3)(2)=12b=2(3)(2)=12;
c=32+22=9+4=13c=32+22=9+4=13 .
6. What is Fermat's Last Theorem?
A. There are infinitely many prime numbers
B. xn+yn=znxn+yn=zn has no positive integer solutions for n>2n>2
C. Every even number is the sum of two primes
D. 22 is irrational
Correct Answer: B
Rationale: Fermat's Last Theorem states that xn+yn=znxn+yn=zn has no positive
integer solutions for n>2n>2 .
7. Which of the following is a primitive Pythagorean triple?
A. (6, 8, 10)
B. (9, 12, 15)
C. (8, 15, 17)
D. (10, 24, 26)
, Correct Answer: C
Rationale: (8, 15, 17) has gcd(8,15,17)=1gcd(8,15,17)=1. The others are multiples
of smaller triples .
8. For hypotenuse c=65c=65, two primitive Pythagorean triples exist. They are:
A. (16, 63, 65) and (33, 56, 65)
B. (25, 60, 65) and (39, 52, 65)
C. (20, 63, 65) and (33, 56, 65)
D. (16, 63, 65) and (25, 60, 65)
Correct Answer: A
Rationale: (16, 63, 65) comes from m=8,n=1m=8,n=1. (33, 56, 65) comes from
m=7,n=4m=7,n=4. (25, 60, 65) is not primitive .
9. In a primitive Pythagorean triple, c=m2+n2c=m2+n2 is always:
A. Even
B. Odd
C. Divisible by 3
D. A perfect square
Correct Answer: B
Rationale: If mm and nn have opposite parity, m2m2 and n2n2 are one even, one odd,
so their sum is odd .
10. The formula a=m2−n2,b=2mn,c=m2+n2a=m2−n2,b=2mn,c=m2+n2
generates all primitive Pythagorean triples when:
