The least non-negative remainder when $2^{75}$ is divided by 5 will be:-

When powers of 2 are divided by 5, there's a repeating pattern in the remainders.

Calculate the first few powers of 2 and find their remainders:

Power	Calculation	Remainder when ÷ 5
2¹	2	2
2²	4	4
2³	8	3 (because 8 = 5×1 + 3)
2⁴	16	1 (because 16 = 5×3 + 1)
2⁵	32	2 (because 32 = 5×6 + 2)

The remainder pattern is: 2, 4, 3, 1 and then it repeats.

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The pattern repeats every 4 powers. The cycle length is 4. So:

• $2^1, 2^5, 2^9, 2^{13}$... all give remainder 2
• $2^2, 2^6, 2^{10}, 2^{14}$... all give remainder 4
• $2^3, 2^7, 2^{11}, 2^{15}$... all give remainder 3
• $2^4, 2^8, 2^{12}, 2^{16}$... all give remainder 1

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Divide the exponent 75 by the cycle length 4:

$75 \div 4 = 18$ remainder $3$

This means:

$75 = 4 \times 18 + 3$

After completing 18 full cycles, we're at the 3rd position in the pattern.

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Looking at the pattern:

• Position 1 → remainder 2
• Position 2 → remainder 4
• Position 3 → remainder 3
• Position 4 → remainder 1

Since 75 leaves remainder 3 when divided by 4, the answer is at the 3rd position in the pattern.

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$2^{75}$

$= 2^{4 \times 18 + 3}$

$= (2^4)^{18} \times 2^3$

$= (16)^{18} \times 8$

Since 16 leaves remainder 1 when divided by 5, $(16)^{18}$ also leaves remainder 1.

And 8 leaves remainder 3 when divided by 5.

Therefore: $1 \times 3 = 3$

The least non-negative remainder when $2^{75}$ is divided by 5 is 3.