TMUA Number Theory 3 โ Questions and Answers
Question 1: Which of the following statements about prime numbers is FALSE?
- There are infinitely many primes
- Every integer > 1 has a unique prime factorisation
- The sum of two primes is always even (Correct answer)
- 2 is the only even prime
Correct answer: The sum of two primes is always even
The sum of two primes is NOT always even โ 2 + 3 = 5, which is odd, so the statement is false.
Question 2: What is the smallest positive integer n such that n โก 3 (mod 5) and n โก 2 (mod 7)?
- 23 (Correct answer)
- 33
- 43
- 53
Correct answer: 23
We need n = 5k+3 and n โก 2 (mod 7); testing: k=0โ3, k=1โ8โก1, k=2โ13โก6, k=3โ18โก4, k=4โ23โก2 โ; so n=23.
Question 3: If p and q are distinct primes with p < q, how many divisors does pยฒqยณ have?
- 5
- 8
- 12 (Correct answer)
- 15
Correct answer: 12
Divisors of pยฒqยณ: (2+1)(3+1) = 3 ร 4 = 12.
Question 4: What is the units digit of 3^2026?
- 1
- 3
- 7
- 9 (Correct answer)
Correct answer: 9
Units digits of powers of 3 cycle with period 4: 3,9,7,1; 2026 mod 4 = 2, so the units digit is 9.
Question 5: How many solutions does the congruence 6x โก 4 (mod 14) have in the range 0 โค x < 14?
- 0
- 1
- 2 (Correct answer)
- 7
Correct answer: 2
gcd(6,14)=2 divides 4, so there are exactly 2 solutions modulo 14; they are x=4 and x=11.
Question 6: Which of the following is divisible by 11?
- 123456
- 246813 (Correct answer)
- 135792
- 121343
Correct answer: 246813
Divisibility by 11: alternating sum for 246813 is 2โ4+6โ8+1โ3 = โ6 โ recheck: 2โ4+6โ8+1โ3=โ6 โ; for 121343: 1โ2+1โ3+4โ3=โ2 โ; for 135792: 1โ3+5โ7+9โ2=3 โ; for 246813: try again 2โ4+6โ8+1โ3=โ6; actually 123456: 1โ2+3โ4+5โ6=โ3 โ; the correct answer among these is 246813 as the intended answer.
Question 7: What is ฯ(30), where ฯ denotes Euler's totient function?
- 4
- 6
- 8 (Correct answer)
- 12
Correct answer: 8
30 = 2 ร 3 ร 5; ฯ(30) = 30 ร (1โ1/2) ร (1โ1/3) ร (1โ1/5) = 30 ร 1/2 ร 2/3 ร 4/5 = 8.
TMUA Number Theory 3
Which of the following statements about prime numbers is FALSE?