Question

Unit digit in 2487power395

Answer 1

If a number is divided by 10, then the remainder obtained will be equal to the ones digit of the number.

10x + y  leaves remainder y.  

Here, the ones digit of 2487 is 7, so 2487 divided by 10 leaves remainder 7. This can be written as,

2487 ≡ 7 (mod 10)   →   (1)

Now, take the 4th power of both sides.

2487⁴ ≡ 7⁴ (mod 10)

2487⁴ ≡ 2401 (mod 10)

2401 divided by 10 leaves remainder 1, the ones digit of 2401.

2401 ≡ 1 (mod 10)

Thus,

2487⁴ ≡ 1 (mod 10)

Remainder 1 is got. Now we can find the answer.

First we have to divide the exponent in the question 395 by 4.

395 divided by 4 gives quotient 98 and remainder 3.

Thus, take the 98th power of both sides of 2487⁴ ≡ 1 (mod 10).

(2487⁴)⁹⁸ ≡ 1⁹⁸ (mod 10)

2487³⁹² ≡ 1 (mod 10)

Now, multiply 2487³ to both sides.

2487³⁹² × 2487³ ≡ 1 × 2487³ (mod 10)

2487³⁹²⁺³ ≡ 2487³ (mod 10)

2487³⁹⁵ ≡ 2487³ (mod 10)   →   (2)

Now, from (1),

2487 ≡ 7 (mod 10)

Cubing both sides,

2487³ ≡ 7³ (mod 10)

2487³ ≡ 343 (mod 10)

As 343 divided by 10 leaves remainder 3, which is 343 ≡ 3 (mod 10),

2487³ ≡ 3 (mod 10)

Thus, from (2),

2487³⁹⁵ ≡ 2487³ (mod 10)

2487³⁹⁵ ≡ 3 (mod 10)

So the ones digit of 2487³⁹⁵ is 3.