If x is the remainder when 3⁶¹²⁸⁴ is divided by 5 and y is the remainder when 4⁹⁶ is divided by 6, then what the value of (2x - y)?

Solution (By Examveda Team)

x is the remainder when 3⁶¹²⁸⁴ is divided by 5
So, $$\frac{{{3^{61284}}}}{5} = \frac{{{3^{4 \times 15321}}}}{5} = \frac{{{3^4}}}{5} = \frac{{81}}{5}$$
Remainder = 1
x = 1
y is the remainder when 4⁹⁶ is divided by 6
$$\frac{{{4^{96}}}}{6} = \frac{{{4^{4 \times 24}}}}{6} = \frac{{{4^4}}}{6} = \frac{{256}}{6}$$
Remainder = 4
y = 4
Now, (2x - y)
= 2 - 4
= -2

Alternate solution:
As we know,
Unit digit of 3⁶¹²⁸⁴
⇒ 3⁴
⇒ 1
So we can say if we divide 3⁶¹²⁸⁴ by 5 then we get remainder 1
x = 1
As we know,
4 ÷ 6, then remainder 4
4² ÷ 6, then remainder 4
4³ ÷ 6, then remainder 4
4ⁿ ÷ 6, then remainder 4
When n = natural number
So we can say 4⁹⁶ is divided by 6, then the remainder is 4
y = 4
Now, (2x - y)
⇒ 2 × 1 - 4
⇒ 2 - 4
⇒ -2