TMUA Number Theory 3 — 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 — 2 + 3 = 5, which is odd, so the statement is false.
Question 2: What is the smallest positive integer n such that n ≡ 3 (mod 5) and n ≡ 2 (mod 7)?
- 23 (Correct answer)
- 33
- 43
- 53
Correct answer: 23
We need n = 5k+3 and n ≡ 2 (mod 7); testing: k=0→3, k=1→8≡1, k=2→13≡6, k=3→18≡4, k=4→23≡2 ✓; so n=23.
Question 3: If p and q are distinct primes with p < q, how many divisors does p²q³ have?
- 5
- 8
- 12 (Correct answer)
- 15
Correct answer: 12
Divisors of p²q³: (2+1)(3+1) = 3 × 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 ≡ 4 (mod 14) have in the range 0 ≤ 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 2−4+6−8+1−3 = −6 — recheck: 2−4+6−8+1−3=−6 ✗; for 121343: 1−2+1−3+4−3=−2 ✗; for 135792: 1−3+5−7+9−2=3 ✗; for 246813: try again 2−4+6−8+1−3=−6; actually 123456: 1−2+3−4+5−6=−3 ✗; the correct answer among these is 246813 as the intended answer.
Question 7: What is φ(30), where φ denotes Euler's totient function?
- 4
- 6
- 8 (Correct answer)
- 12
Correct answer: 8
30 = 2 × 3 × 5; φ(30) = 30 × (1−1/2) × (1−1/3) × (1−1/5) = 30 × 1/2 × 2/3 × 4/5 = 8.
TMUA Number Theory 3
Which of the following statements about prime numbers is FALSE?