TMUA Number Theory 3

Question 1

Which of the following statements about prime numbers is FALSE?

Accepted Answer

The sum of two primes is always even

Answer

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)?

Accepted Answer

23

Answer

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?

Accepted Answer

12

Answer

Divisors of p²q³: (2+1)(3+1) = 3 × 4 = 12.

Question 4

What is the units digit of 3^2026?

Accepted Answer

9

Answer

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?

Accepted Answer

2

Answer

gcd(6,14)=2 divides 4, so there are exactly 2 solutions modulo 14; they are x=4 and x=11.

(TMUA) Test of Mathematics for University Admission Practice Test