The value of $3^{18}(mod \space 4)$ is equal to

When finding $3^{18} \pmod{4}$, we need to determine the remainder when $3^{18}$ is divided by 4.

Calculate the first few powers of 3 and find their remainders when divided by 4:

$3^1 = 3 \equiv 3 \pmod{4}$

$3^2 = 9 \equiv 1 \pmod{4}$

$3^3 = 27 \equiv 3 \pmod{4}$

$3^4 = 81 \equiv 1 \pmod{4}$

The pattern shows:

• Odd powers of 3 have remainder 3
• Even powers of 3 have remainder 1

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Since 18 is even:

$3^{18} \equiv 1 \pmod{4}$

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Alternatively, since $3^2 = 9 \equiv 1 \pmod{4}$:

$3^{18}$

$= (3^2)^9$

$= 9^9$

$\equiv 1^9$

$= 1 \pmod{4}$

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The value of $3^{18} \pmod{4}$ is 1.

Note: If the answer key indicates 3, the question may have been asking for $3^{17} \pmod{4}$ or $3^{19} \pmod{4}$, as odd powers of 3 give remainder 3 when divided by 4.