Math / Numbers & Operations

QuestionThe remainder obtained when $17^{201}$ is divided by 18 , is :

Studdy Solution

STEP 1

1. We are looking for the remainder when $17^{201}$ is divided by 18.
2. We can use properties of modular arithmetic to simplify the problem.

STEP 2

1. Simplify the base using modular arithmetic.
2. Use properties of exponents in modular arithmetic.
3. Calculate the remainder.

STEP 3

Simplify the base $17$ modulo $18$ .
$17 \equiv -1 \pmod{18}$

STEP 4

Use the property of exponents in modular arithmetic:
$17^{201} \equiv (-1)^{201} \pmod{18}$

STEP 5

Calculate $(-1)^{201}$ .
Since 201 is an odd number:
$(-1)^{201} = -1$
Thus:
$(-1)^{201} \equiv -1 \pmod{18}$

STEP 6

Convert $-1$ to a positive remainder:
$-1 \equiv 17 \pmod{18}$
The remainder when $17^{201}$ is divided by 18 is:
$\boxed{17}$

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QuestionThe remainder obtained when $17^{201}$ is divided by 18 , is :

Studdy Solution

STEP 1

1. We are looking for the remainder when $17^{201}$ is divided by 18.
2. We can use properties of modular arithmetic to simplify the problem.

STEP 2

1. Simplify the base using modular arithmetic.
2. Use properties of exponents in modular arithmetic.
3. Calculate the remainder.

STEP 3

Simplify the base $17$ modulo $18$ .
$17 \equiv -1 \pmod{18}$

STEP 4

Use the property of exponents in modular arithmetic:
$17^{201} \equiv (-1)^{201} \pmod{18}$

STEP 5

Calculate $(-1)^{201}$ .
Since 201 is an odd number:
$(-1)^{201} = -1$
Thus:
$(-1)^{201} \equiv -1 \pmod{18}$

STEP 6

Convert $-1$ to a positive remainder:
$-1 \equiv 17 \pmod{18}$
The remainder when $17^{201}$ is divided by 18 is:
$\boxed{17}$

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QuestionThe remainder obtained when 1720117^{201}17201 is divided by 18 , is :

Studdy Solution

STEP 1

1. We are looking for the remainder when 17201 17^{201} 17201 is divided by 18.2. We can use properties of modular arithmetic to simplify the problem.

STEP 2

1. Simplify the base using modular arithmetic.2. Use properties of exponents in modular arithmetic.3. Calculate the remainder.

STEP 3

Simplify the base 17 17 17 modulo 18 18 18.17≡−1(mod18) 17 \equiv -1 \pmod{18} 17≡−1(mod18)

STEP 4

Use the property of exponents in modular arithmetic: 17201≡(−1)201(mod18) 17^{201} \equiv (-1)^{201} \pmod{18} 17201≡(−1)201(mod18)

STEP 5

Calculate (−1)201 (-1)^{201} (−1)201.Since 201 is an odd number:(−1)201=−1 (-1)^{201} = -1 (−1)201=−1Thus:(−1)201≡−1(mod18) (-1)^{201} \equiv -1 \pmod{18} (−1)201≡−1(mod18)

STEP 6

Convert −1-1−1 to a positive remainder:−1≡17(mod18) -1 \equiv 17 \pmod{18} −1≡17(mod18) The remainder when 17201 17^{201} 17201 is divided by 18 is:17\boxed{17}17​

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QuestionThe remainder obtained when $17^{201}$ is divided by 18 , is :

1. We are looking for the remainder when $17^{201}$ is divided by 18.
2. We can use properties of modular arithmetic to simplify the problem.

1. Simplify the base using modular arithmetic.
2. Use properties of exponents in modular arithmetic.
3. Calculate the remainder.

Simplify the base $17$ modulo $18$ .
$17 \equiv -1 \pmod{18}$

Use the property of exponents in modular arithmetic:
$17^{201} \equiv (-1)^{201} \pmod{18}$

Calculate $(-1)^{201}$ .
Since 201 is an odd number:
$(-1)^{201} = -1$
Thus:
$(-1)^{201} \equiv -1 \pmod{18}$

Convert $-1$ to a positive remainder:
$-1 \equiv 17 \pmod{18}$
The remainder when $17^{201}$ is divided by 18 is:
$\boxed{17}$