Remainder of a to the power n, divided by b. | Algebra - Number Theory

Answer: 1

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Explanation :

Let us find the pattern that remainders follow when successive powers of 7 are divided by 8.

Remainder of 7¹/8 = 7.
Remainder of 7²/8 = 1.
Remainder of 7³/8 = 7.
Remainder of 7⁴/8 = 1.

∴ We find that the remainders are repeated after every two powers. 

∴ Remainder of 7⁶⁴ when divided by 8 is the same as 7² when divided by 8, since 64 is a multiple of 2. 

⇒ The remainder is 1.

Alternately,

$R\left[\frac{7^{64}}{8}\right]$ = ${\left[R\left(\frac{7}{8}\right)\right]}^{64}$ = [-1]⁶⁴ = 1

Hence, 1.

Answer: 2

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Explanation :

Let us find the pattern that remainders follow when successive powers of 2 are divided by 7.

Remainder of 2¹/7 = 2.
Remainder of 2²/7 = 4.
Remainder of 2³/7 = 1.
Remainder of 2⁴/7 = 2.
Remainder of 2⁵/7 = 4.
Remainder of 2⁶/7 = 1.

∴ We find that the remainders are repeated after every three powers. 

∴ $R\left[\frac{2^{100}}{7}\right]$ =  $R\left[\frac{2^{99}\times 2^1}{7}\right]$ = $R\left[\frac{2^{99}}{7}\right]\times R\left[\frac{2^1}{7}\right]$ = 1 × 2 = 2

⇒ The remainder is 2.

Hence, 2.

Answer: 4

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Explanation :

Let us find the pattern that remainders follow when successive powers of 3 are divided by 7.

Remainder of 3¹/7 = 3.
Remainder of 3²/7 = 2.
Remainder of 3³/7 = 6.
Remainder of 3⁴/7 = 4.
Remainder of 3⁵/7 = 5.
Remainder of 3⁶/7 = 1.
Remainder of 3⁷/7 = 3.
Remainder of 3⁸/7 = 2.

∴ We find that the remainders are repeated after every six powers. 

∴ $R\left[\frac{3^{100}}{7}\right]$ =  $R\left[\frac{3^{96}\times 3^4}{7}\right]$ = $R\left[\frac{3^{96}}{7}\right]\times R\left[\frac{3^4}{7}\right]$ = 1 × 4 = 4.

⇒ The remainder is 4.

Alternately,

Let us find the pattern that remainders follow when successive powers of 3 are divided by 7.

Remainder of 3¹/7 = 3.
Remainder of 3²/7 = 2.
Remainder of 3³/7 = 6 = -1. (Concept of negative remainder)

∴ R(3³/7) = -1

∴ $R\left[\frac{3^{100}}{7}\right]$=  $R\left[\frac{3^{99}\times 3^1}{7}\right]$ = $R\left[\frac{3^{99}}{7}\right]\times R\left[\frac{3^1}{7}\right]$ = ${\left(R\left[\frac{3^3}{7}\right]\right)}^{33}$ × 3 = (-1)³³ × 3 = -3 ≡ 7 - 3 = 4.

Hence, 4.

Answer: 3

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Explanation :

Let us find the pattern that remainders follow when successive powers of 3 are divided by 7.

Remainder when 7¹/10 = 7.
Remainder when 7²/10 = 9.
Remainder when 7³/10 = 3.
Remainder when 7⁴/10 = 1.
Remainder when 7⁵/10 = 7.
Remainder when 7⁶/10 = 9.

∴ We find that the remainders are repeated after every four powers. 

$R\left[\frac{7^{83}}{10}\right]$ = $R\left[\frac{7^{80}\times 7^3}{10}\right]$ = $R\left[\frac{7^{80}}{10}\right]$ × $R\left[\frac{7^3}{10}\right]$ = 1 × 3 = 3.

Alternately,

Remainder of any number when divided by 10 is same as the last digit of that number.

∴ We have to find the last digit of 7⁸³.

Last of 7⁸³ = last digit of  7⁸⁰ × 7³ (Cyclicity of last digit of powers of 7 is 4.)

= last digit of 7³

= 3

Hence, 3.

Answer: 0

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Explanation :

There is no need to find the pattern of remainders here.

Divisor, 64 = 2⁶.

Power of 2 in 6³³ is definitely greater than 6.

∴ 6³³ is perfectly divisible by 64 i.e., remainder is 0.

Hence, 0.

Answer: 1

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Explanation :

Let us find the pattern that remainders follow when successive powers of 2 are divided by 17.

Remainder of 2¹/17 = 2.
Remainder of 2²/17 = 4.
Remainder of 2³/17 = 8.
Remainder of 2⁴/17 = 16 = -1.

∴ Remainder will repeat after every 8 terms. 

or, R[2⁴/17] = -1

⇒ R[2¹²⁰/7] = R[2^4×30/7] = R[2⁴/7]³⁰ = (-1)³⁰ = 1.

Hence, 1.

Answer: 4

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Explanation :

R[19¹²²/17] = R[19/17]¹²² = R[2¹²²/17]

Let us find the pattern that remainders follow when successive powers of 2 are divided by 17.

Remainder of 2¹/17 = 2.
Remainder of 2²/17 = 4.
Remainder of 2³/17 = 8.
Remainder of 2⁴/17 = 16 = -1.

∴ Remainder will repeat after every 8 terms. 

or, R[2⁴/17] = -1

⇒ R[2¹²²/7] = R[2^4×30+2/7] = R[2¹²⁰/7] × R[2²/7] = R[2⁴/7]³⁰ × R[2²/7] = (-1)³⁰ × 4 = 4.

Hence, 4.