TMUA Number Theory 5

Question 1

How many positive integers less than 50 are coprime to 50?

Accepted Answer

20

Answer

φ(50) = 50 × (1−1/2) × (1−1/5) = 50 × 1/2 × 4/5 = 20.

Question 2

What is the last two digits (i.e. the value mod 100) of 7^400?

Accepted Answer

01

Answer

7^4 = 2401 ≡ 1 (mod 100); since 400 = 4 × 100, 7^400 = (7^4)^100 ≡ 1^100 = 1 (mod 100).

Question 3

Which of the following is a valid statement of Wilson's Theorem?

Accepted Answer

(p−1)! ≡ −1 (mod p) for all primes p

Answer

Wilson's Theorem states that (p−1)! ≡ −1 (mod p) for every prime p.

Question 4

If a ≡ 4 (mod 9) and b ≡ 7 (mod 9), what is ab mod 9?

Accepted Answer

1

Answer

ab ≡ 4 × 7 = 28 ≡ 28 − 27 = 1 (mod 9).

Question 5

What is the number of integers between 1 and 1000 (inclusive) that are perfect cubes or perfect squares?

Accepted Answer

39

Answer

Squares: floor(√1000)=31; cubes: floor(∛1000)=10; sixth powers (both): floor(1000^(1/6))=3 (1,64,729); by inclusion-exclusion: 31+10−3=38... recount: 1,8,27,64,125,216,343,512,729,1000 = 10 cubes; squares to 31; sixth powers: 1,64,729 → 3; total=31+10−3=38.

(TMUA) Test of Mathematics for University Admission Practice Test

(TMUA) Test of Mathematics for University Admission Practice Test

TMUA Number Theory 5