Question

What is the remainder when $(2^{55}+3^{23})$ is divided by 5?

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer:

A. 0

Shortcut Trick

$(2^{55}+3^{23})$

Power of 2 = $55\div{4}$ = Remainder is 3 

Power of 3 = $23\div{4}$ = Remainder is 3 

Then,

(2³ + 3³) ÷ 5

⇒ (8 + 27) ÷ 5 

⇒ 35 ÷ 5 = 7, which means here remainder is 0.

Alternate Method

We can write, 2⁵⁵ = (2⁴)¹³ × 2³

⇒ Unit digit (2⁵⁵) = Unit digit (16¹³ $\times$ 8)

⇒ Unit digit (255) = Unit digit (613 $\times$ 8)

⇒ Unit digit (2⁵⁵) = Unit digit (6 $\times$ 8) = 8

Similarly

,

We can write, 3²³ = (3⁴)⁵ $\times$ 3³

⇒ Unit digit (3²³) = Unit digit (81⁵ $\times$ 7)

⇒ Unit digit (323) = Unit digit (15 $\times$ 7)

⇒ Unit digit (3²³) = 7

Now,

Unit digit (2⁵⁵ + 3²³) = Unit digit (8 + 7) = 5

∵ When 5 is divided by 5, the remainder is 0

∴ When (2⁵⁵ + 3²³) is divided by 5, the remainder is 0.