Determine the positive remainder when the value of 36^{43} is divided by 37. — State PSC Exam

Determine the positive remainder when the value of $36^{43}$ is divided by $37$ .

Answer: 36

Answer

By the properties of modular arithmetic,

36

is congruent to

-1

modulo

37

. When we raise both sides to the power of

43

, we obtain

36^{43} \equiv (-1)^{43} \pmod{37}

. Since an odd power of

-1

is still

-1

, the expression leaves a remainder of

-1

. To find the standard positive remainder, we simply add the divisor to this result:

-1 + 37 = 36

Step-by-Step Solution

Express the base as a negative remainder modulo the divisor.

When

36

is divided by

37

, the remainder can be expressed as

-1

because

36 = 37 \times 1 - 1

. Therefore, we can state that

36 \equiv -1 \pmod{37}

Using a negative remainder with an absolute value of

1

makes calculating large exponentiations trivial.

Apply the exponent to the negative remainder and convert back to a positive value.

Calculate

(-1)^{43}

. Since

43

is an odd number,

(-1)^{43} = -1

. To find the valid positive remainder, add the divisor:

-1 + 37 = 36

The rules of modular arithmetic allow us to raise the remainder to the given power. However, the final answer must always be a non-negative integer strictly less than the divisor.

Key Concept

Negative Remainders for Large Powers

Alternative Method

You can use Fermat's Little Theorem, which states

a^{p-1} \equiv 1 \pmod p

for any prime

p

(as long as

a

is not divisible by

p

). Here,

p = 37

, so

36^{36} \equiv 1 \pmod{37}

. We can split the expression:

36^{43} = 36^{36} \times 36^7 \equiv 1 \times 36^7 \pmod{37}

. Using the negative remainder trick for the remaining part gives

(-1)^7 = -1

. Adding

37

yields the positive remainder of

36

Estimated Time:30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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