**Question 1:**

What is the remainder when the product 1998 × 1999 × 2000 is divided by 7?

[1] 0

[2] 1

[3] 2

[4] 4

**Answer & Solution**

The remainders when 1998, 1999, and 2000 are divided by 7 are 3, 4, and 5 respectively. Hence the final remainder is the remainder when the product 3 × 4 × 5 = 60 is divided by 7. Therefore, remainder = 4

**Question 2:**

What is the remainder when 2

^{2004}is divided by 7?

[1] 4

[2] 1

[3] 2

[4] 6

**Answer & Solution**

2^{2004} is again a product (2 × 2 × 2... (2004 times)). Since 2 is a number less than 7 we try to convert the product into product of numbers higher than 7. Notice that 8 = 2 × 2 × 2. Therefore we convert the product in the following manner- 2^{2004} = 8^{668} = 8 × 8 × 8... (668 times).

The remainder when 8 is divided by 7 is 1. Hence the remainder when 8^{668} is divided by 7 is the remainder obtained when the product 1 × 1 × 1... is divided by 7. Therefore, remainder = 1

**Question 3:**

What is the remainder when 2

^{2006}is divided by 7?

**Answer & Solution**

This problem is like the previous one, except that 2006 is not an exact multiple of 3 so we cannot convert it completely into the form 8^{x}. We will write it in following manner- 2^{2006} = 8^{668} × 4.

Now, 8^{668} gives the remainder 1 when divided by 7 as we have seen in the previous problem. And 4 gives a remainder of 4 only when divided by 7. Hence the remainder when 2^{2006} is divided by 7 is the remainder when the product 1 × 4 is divided by 7. Therefore, remainder = 4

**Question 4:**

What is the remainder when 25

^{25}is divided by 9?

**Answer & Solution**

Again 25^{25} = (18 + 7)^{25} = (18 + 7)(18 + 7)...25 times = 18K + 7^{25}

Hence remainder when 25^{25} is divided by 9 is the remainder when 7^{25} is divided by 9.

Now 7^{25} = 7^{3} × 7^{3} × 7^{3}.. (8 times) × 7 = 343 × 343 × 343... (8 times) × 7.

The remainder when 343 is divided by 9 is 1 and the remainder when 7 is divided by 9 is 7.

Hence the remainder when 7^{25} is divided by 9 is the remainder we obtain when the product 1 × 1 × 1... (8 times) × 7 is divided by 9. The remainder is 7 in this case. Hence the remainder when 25^{25} is divided by 9 is 7.

**Question 5:**

What the remainder when 2

^{96}is divided by 96?

**Answer & Solution**

The common factor between 2^{96} and 96 is 32 = 2^{5}.

Removing 32 from the dividend and the divisor we get the numbers 2^{91} and 3 respectively.

The remainder when 2^{91} is divided by 3 is 2.

Hence the real remainder will be 2 multiplied by common factor 32.

Remainder = 64

**Question 6:**

Find the remainder when 7

^{52}is divided by 2402.

**Answer & Solution**

7^{52} = (7^{4})^{13} = (2401)^{13} = (2402 – 1)^{13} = 2402K + (−1)^{13} = 2402K −1.

Hence, the remainder when 7^{52} is divided by 2402 is equal to −1 or 2402 – 1 = 2401.

Remainder = 2401.

**Question 7:**

What is the remainder when 3

^{444}+ 4

^{333}is divided by 5?

**Answer & Solution**

The dividend is in the form a^{x} + b^{y}. We need to change it into the form a^{n} + b^{n}.

3^{444} + 4^{333} = (3^{4})^{111} + (4^{3})^{111}. Now (3^{4})^{111} + (4^{3})^{111} will be divisible by 3^{4} + 4^{3} = 81 + 64 = 145. Since the number is divisible by 145 it will certainly be divisible by 5. Hence, the remainder is 0.

**Question 8:**

What is the remainder when (5555)

^{2222}+ (2222)

^{5555}is divided by 7?

**Answer & Solution**

The remainders when 5555 and 2222 are divided by 7 are 4 and 3 respectively. Hence, the problem reduces to finding the remainder when (4)^{2222} + (3)^{5555} is divided by 7.

Now (4)^{2222} + (3)^{5555} = (4^{2})^{1111} + (3^{5})^{1111} = (16)^{1111} + (243)^{1111}. Now (16)^{1111} + (243)^{1111} is divisible by 16 + 243 or it is divisible by 259, which is a multiple of 7. Hence the remainder when (5555)^{2222} + (2222)^{5555} is divided by 7 is zero.

**Question 9:**

20

^{2004}+ 16

^{2004}– 3

^{2004}− 1 is divisible by:

[1] 317

[2] 323

[3] 253

[4] 91

**Answer & Solution**

20^{2004} + 16^{2004} – 3^{2004} – 1 = (20^{2004} – 3^{2004}) + (16^{2004} – 1^{2004}). Now 20^{2004} – 3^{2004} is divisible by 17 (Theorem 3) and 16^{2004} – 1^{2004} is divisible by 17 (Theorem 2). Hence the complete expression is divisible by 17.

20^{2004} + 16^{2004} – 3^{2004} – 1 = (20^{2004} – 1^{2004}) + (16^{2004} – 3^{2004}). Now 20^{2004} – 1^{2004} is divisible by 19 (Theorem 3) and 16^{2004} – 3^{2004} is divisible by 19 (Theorem 2). Hence the complete expression is also divisible by 19.

Hence the complete expression is divisible by 17 × 19 = 323.

**Question 10:**

Find the remainder when 5

^{37}is divided by 63.

**Answer & Solution**

5 and 63 are coprime to each other, therefore we can apply Euler’s theorem here.

63 = 3^{2} × 7 $\Rightarrow$ $\phi (63)=63(1-\frac{1}{3})(1-\frac{1}{7})=36$

Therefore, Remainder $[\frac{{{5}^{37}}}{63}]$

=Remainder $[\frac{{{5}^{36}}\times 5}{63}]=5$

**Question 11:**

What is the remainder when n

^{7}– n is divided by 42?

**Answer & Solution**

Since 7 is prime, n^{7} – n is divisible by 7. n^{7} – n = n(n^{6} – 1) = n (n + 1)(n – 1)(n^{4} + n^{2} + 1). Now (n – 1)(n)(n + 1) is divisible by 3! = 6. Hence n^{7} – n is divisible by 6 x 7 = 42. Hence the remainder is 0.

**Question 12:**

Find the remainder when 52

^{60}is divided by 31.

**Answer & Solution**

31 is a prime number therefore f(N) = 30. 52 and 31 are prime to each other. Therefore, by Fermat’s theorem:

Remainder $[\frac{{{52}^{30}}}{31}]=1$

$\Rightarrow$ Remainder $[\frac{{{52}^{60}}}{31}]=1$

**Question 13:**

Find the remainder when 40! is divided by 41.

**Answer & Solution**

By Wilson’s theorem, we can see that 40! + 1 is divisible by 41

$\Rightarrow$ Remainder$[\frac{40!}{41}]=41-1=40$

**Question 14:**

Find the remainder when 39! is divided by 41.

**Answer & Solution**

In the above problem, we saw that the remainder when 40! is divided by 41 is 40.

$\Rightarrow$ 40! = 41k + 40 $\Rightarrow$ 40 × 39! = 41k + 40. The R.H.S. gives remainder 40 with 41 therefore L.H.S. should also give remainder 40 with 41. L.H.S. = 40 × 39! where 40 gives remainder 40 with 41. Therefore, 39! should give remainder 1 with 41.

**Question 15:**

Find the remainder when 3

^{101}is divided by 77.

**Answer & Solution**

77 = 11 × 7.

By Fermat’s little theorem, Remainder $\left[ \frac{{{3}^{6}}}{7} \right]=1$ and Remainder $\left[ \frac{{{3}^{10}}}{11} \right]=1$

Remainder $\left[ \frac { 3 ^ { 101 } } { 7 } \right]$=Remainder $\left[ \frac { 3 ^ { 96 } \times 3 ^ { 5 } } { 7 } \right]$=Remainder $\left[ \frac { \left( 3 ^ { 6 } \right) ^ { 16 } \times 3 ^ { 5 } } { 7 } \right]$=Remainder $\left[ \frac { 1 \times 3 ^ { 5 } } { 7 } \right] = 5 = \mathrm { r } _ { 1 }$

Remainder $\left[ \frac { 3 ^ { 101 } } { 11 } \right]$=Remainder $\left[ \frac { 3 ^ { 100 } \times 3 } { 11 } \right]$=Remainder $\left[ \frac { \left( 3 ^ { 10 } \right) ^ { 10 } \times 3 } { 11 } \right]$=Remainder $\left[ \frac { 1 \times 3 } { 11 } \right] = 3 = \mathrm { r } _ { 2 }$

Now we will find x and y such that 7x + 11y = 1. By observation we can find out, x = -3 and y = 2.

Now we can say that Remainder $\left[ \frac{{{3}^{101}}}{77} \right]=7\times 3\times -3+11\times 5\times 2=47$

**Question 16:**

Find the remainder when 3

^{101}is divided by 77.

**Answer & Solution**

$\phi \left( 77 \right)=77\left( 1-\frac{1}{7} \right)\left( 1-\frac{1}{11} \right)=60$

Remainder $\left[ \frac { 60 } { 77 } \right] = 1$

$\Rightarrow$ Remainder $\left[ \frac { 3 ^ { 101 } } { 77 } \right] =$ Remainder $\left[ \frac{{{3}^{60}}\times {{3}^{41}}}{77} \right]$

=Remainder $\left[ \frac { 1 \times 3 ^ { 41 } } { 77 } \right] =$ Remainder$\left[ \frac { 3 ^ { 41 } } { 77 } \right]$

Remainder $\left[ \frac { 3 ^ { 4 } } { 77 } \right] =$ Remainder $\left[ \frac { 81 } { 77 } \right] = 4$

$\Rightarrow$ Remainder $\left[ \frac { 3 ^ { 41 } } { 77 } \right] =$ Remainder $\left[ \frac { \left( 3 ^ { 4 } \right) ^ { 10 } \times 3 } { 77 } \right]$= Remainder $\left[ \frac{{{4}^{10}}\times 3}{77} \right]$

= Remainder $\left[ \frac { 4 ^ { 4 } \times 4 ^ { 4 } \times 4 ^ { 2 } \times 3 } { 77 } \right] =$ Remainder $\left[ \frac { 256 \times 256 \times 48 } { 77 } \right]$

= Remainder $\left[ \frac { 25 \times 25 \times 48 } { 77 } \right] =$ Remainder $\left[ \frac { 9 \times 48 } { 77 } \right] = 47$

**Question 17:**

Find the remainder when${{32}^{{{32}^{32}}}}$is divided by 9.

**Answer & Solution**

Notice that 32 and 9 are coprime. $\phi (9)=9\left( 1-\frac{1}{3} \right)=6$

Hence by Euler’s theorem, Remiander$\left[ \frac{{{32}^{6}}}{9} \right]=1$. Since the power is 32^{32}, we will have to simplify this power in terms of 6k + r. Therefore, we need to find the remainder when 32^{32} is divided by 6.

Remainder $\left[ \frac { 32 ^ { 32 } } { 6 } \right] =$ Remainder $\left[ \frac { 2 ^ { 32 } } { 6 } \right]$

= Remainder $\left[ \frac { \left( 2 ^ { 8 } \right) ^ { 4 } } { 6 } \right] =$ Remainder $\left[ \frac { 256 \times 256 \times 256 \times 256 } { 6 } \right]$

= Remainder $\left[ \frac { 256 } { 6 } \right] = 4$.

Therefore, ${{32}^{{{32}^{32}}}}$= 32^{6k + 4 }= (32^{6})^{k} × 32^{4}

$\Rightarrow$ Remainder $\left[ \frac { \left( 32 ^ { 6 } \right) ^ { \mathrm { k } } \times 32 ^ { 4 } } { 9 } \right] =$ Remainder $\left[ \frac { 32 ^ { 4 } } { 9 } \right]$

=Remainder $\left[ \frac { 5 \times 5 \times 5 \times 5 } { 9 } \right] =$ Remainder $\left[ \frac { 625 } { 9 } \right] = 4$

**Question 18:**

What will be the remainder when N = 10

^{10}+ 10

^{100}+ 10

^{1000}+........... + 10

^{10000000000}is divided by 7?

**Answer & Solution**

By Fermat’s Little Theorem 10^{6} will give remainder as 1 with 7.

Remainder $\left[ \frac { 10 ^ { 10 } } { 7 } \right] =$ Remainder $\left[ \frac { 10 ^ { 6 } \times 10 ^ { 4 } } { 7 } \right]$

=Remainder $\left[ \frac { 10 ^ { 4 } } { 7 } \right] =$ Remainder $\left[ \frac { 3 ^ { 4 } } { 7 } \right] = 4$

Similarly, all the other terms give remainder of 4 with 7. Therefore, total remainder = 4 + 4 + 4… (10 times) = 40.

Remainder of 40 with 7 = 5

**Question 19:**

What is the remainder when N = 2222

^{5555}+ 5555

^{2222}is divided by 7?

**Answer & Solution**

2222^{6} will give remainder 1 when divided by 7.

5555 = 6K+5 $\Rightarrow$ 2222^{5555} = 2222^{6k + 5 }

$\Rightarrow $ Remainder $\left[ \frac { 2222 ^ { 5555 } } { 7 } \right]$

= Remainder $\left[ \frac { 2222 ^ { 5 } } { 7 } \right]$

=Remainder $\left[ \frac { 5 } { 7 } \right] = 5$

Also 5555^{6} will give remainder 1 when divided by 7.

5555^{2222} = 5555^{6k + 2}

$\Rightarrow$$Remainder[\frac{{{5555}^{2222}}}{7}]=Remainder[\frac{{{5555}^{2}}}{7}]=Remainder[\frac{{{4}^{2}}}{7}]=2$

So final remainder is (5 + 2) divided by 7 = 0

**Question 20:**

Find the remainder when 8

^{643}is divided by 132.

**Answer & Solution**

Note that here 8 and 132 are not co-prime as HCF (8, 132) = 4 and not 1. Therefore, we cannot apply Euler’s theorem directly.

Remainder $\frac { 8 ^ { 643 } } { 132 } ] =$ Remainder $\left[ \frac { 2 ^ { 1929 } } { 132 } \right] = 4 \times$ Remainder $\left[ \frac { 2 ^ { 1927 } } { 33 } \right]$

Now we can apply Euler’s theorem.

$\phi ( 33 ) = 33 \left( 1 - \frac { 1 } { 3 } \right) \left( 1 - \frac { 1 } { 11 } \right) = 20$

$\Rightarrow$ Remainder $\left[ \frac { 2 ^ { 20 } } { 33 } \right] = 1$

= Remainder $\left[ \frac { 2 ^ { 1927 } } { 33 } \right] =$ Remainder $\left[ \frac { 2 ^ { 7 } } { 33 } \right] = 29$

$\Rightarrow$ Real remainder $= 4 \times 29 = 116$