Calculate the true positive remainder obtained upon dividing the numerical expression 67^{… — State PSC Exam

Calculate the true positive remainder obtained upon dividing the numerical expression $67^{95} - 3 \times 53^{42}$ by $17$ .

Answer: 4

Answer

The correct positive remainder is 4.

By evaluating the expression under modulo

17

, we first simplify the base numbers:

67 \equiv -1 \pmod{17}

and

53 \equiv 2 \pmod{17}

. This reduces the overarching expression to

(-1)^{95} - 3 \times 2^{42}

. The first term trivially becomes

-1

. For the second term, we can utilize the fact that

2^4 = 16 \equiv -1 \pmod{17}

. Therefore,

2^{42}

can be broken down into

(2^4)^{10} \times 2^2 \equiv (-1)^{10} \times 4 \equiv 4 \pmod{17}

. Substituting these simplified values back into the expression yields

-1 - 3(4) = -13

. Because standard remainders must be positive, we adjust the negative result by adding the divisor

17

-13

, which gives the true positive remainder of

4

Step-by-Step Solution

Reduce the base numbers

67

and

53

to smaller equivalent values modulo

17

67 = 17 \times 4 - 1 \implies 67 \equiv -1 \pmod{17}

. And

53 = 17 \times 3 + 2 \implies 53 \equiv 2 \pmod{17}

To drastically simplify large exponentiations by substituting smaller, manageable equivalent bases.

Evaluate the remainder of the first term,

67^{95} \pmod{17}

(-1)^{95} = -1 \pmod{17}

An odd exponent applied to a base of

-1

preserves the negative sign.

Simplify the second term's exponentiation,

2^{42} \pmod{17}

, by identifying a nearby power of

2

that relates to

17

Observe that

2^4 = 16 \equiv -1 \pmod{17}

Finding a power that equals

1

-1

modulo

17

creates a highly efficient shortcut for reducing massive exponents.

Break down

2^{42}

using the established property of

2^4

2^{42} = (2^4)^{10} \times 2^2 \equiv (-1)^{10} \times 4 \equiv 1 \times 4 = 4 \pmod{17}

To substitute the

-1

equivalence and systematically compute the modular value of the second term.

Combine both simplified terms into the original arithmetic expression.

The expression evaluates to

-1 - 3 \times 4 = -1 - 12 = -13 \pmod{17}

To find the overall aggregate modular value of the complete mathematical expression.

Convert the negative result into the equivalent true positive remainder.

-13 + 17 = 4

By definition, a valid remainder must be a non-negative integer strictly less than the divisor.

Key Concept

Modular Arithmetic, Exponent Rules, and Negative Remainders

Estimated Time:3m 0s

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