Question

what is the last digit of 2^15*3^9​

Answer 1

We have to find the ones digit of the product  2¹⁵ × 3⁹.  

Consider  2¹⁵.

2⁴ ≡ 6 (mod 10)

Since  6ⁿ ≡ 6 (mod 10),  we can have  2⁴ⁿ ≡ 6 (mod 10).  So,

⇒  (2⁴)³ = 2¹² ≡ 6 (mod 10)

∴   2¹⁵ = 2¹² × 2³ ≡ 6 × 8 = 48 ≡ 8 (mod 10)

Hence the ones digit of 2¹⁵ is 8.

Consider 3⁹.

We have  3⁴ ≡ 1 (mod 10).

⇒  (3⁴)² = 3⁸ ≡ 1 (mod 10)

⇒  3⁹ = 3⁸ × 3 ≡ 1 × 3 = 3 (mod 10)

Hence the ones digit of 3⁹ is 3.

Now,

2¹⁵ × 3⁹ ≡ 8 × 3 = 24 ≡ 4 (mod 10)

Hence,  4  is the answer.