Solution

The correct answer is Both A and R are correct but R is NOT the correct explanation of A.

Key Points

• Step 1: Order of the group (ℤ₅, ×)
  • Multiplicative group modulo n: ℤ_n* = {a ∈ ℤ_n | gcd(a,n)=1}.
  • Order of ℤ_n* = Euler Totient Function φ(n).
  • Since 5 is prime ⇒ φ(5) = 5 − 1 = 4.
  • ℤ₅* = {1,2,3,4} ⇒ |ℤ₅*| = 4.
• Step 2: Order of the group (ℤ₁₃, ×)
  • 13 is prime ⇒ φ(13) = 13 − 1 = 12.
  • ℤ₁₃* = {1,2,3,…,12} ⇒ |ℤ₁₃*| = 12.
• Step 3: Verify Assertion A
  • Check divisibility: 4 | 12.
  • 12 = 4 × 3 ⇒ divisible.
  • Hence Assertion A is true.
• Step 4: Verify Reason R (Lagrange’s Theorem)
  • If H ≤ G (H is subgroup of G) then:
  • |G| = [G:H] · |H| ⇒ |H| divides |G|.
  • This theorem is mathematically correct.
  • Hence Reason R is true.
• Step 5: Check whether R explains A
  • To apply Lagrange, ℤ₅* must be a subgroup of ℤ₁₃*.
  • But subgroup requires SAME binary operation.
  • Operations differ:
    2×3 ≡ 1 (mod 5)
    2×3 ≡ 6 (mod 13)
  • Hence ℤ₅* ⊄ ℤ₁₃*.
  • Divisibility occurs numerically, not via subgroup relation.

Additional Information

• Mathematical Conclusion:
  • A is TRUE (4 divides 12).
  • R is TRUE (Lagrange theorem).
  • But R cannot justify A because subgroup condition fails.
  • Therefore, option 2 is correct.