Number Systems

• 8
• 8 + 11 = 19
• 19 + 11 = 30
• 30 + 11 = 41

To solve the compound inequality , we will break it into two separate inequalities and solve each one individually.

1. Solve the first inequality:

Add 6 to both sides:

Divide both sides by 8:

1. Solve the second inequality:

Add 6 to both sides:

Divide both sides by 8:

We combine the solutions from both parts:

This represents the interval , which is the correct solution.

To find the remainder when is divided by 7, we can use Fermat's Little Theorem. Fermat's Little Theorem states that if is a prime number and is an integer not divisible by , then .

Here, and . According to the theorem, .

We need to express in terms of a multiple of 6 plus a remainder, since the exponent is large. We perform division: .

Thus, . As per Fermat's Theorem, .

Now, we need to calculate :

. Calculate :

Since with a remainder of 6, .

Therefore, the remainder when is divided by 7 is 6.

To find the least non-negative remainder when is divided by 7, we use Fermat's Little Theorem which states that if is a prime number and is an integer not divisible by , then .

In this case, and , so according to the theorem, .

This means any power of 3 that is a multiple of 6 will have a remainder of 1 when divided by 7.

First, express 51 in terms of multiples of 6:

.

Thus, we can write:

.

Since , we have:

.

Therefore:

.

Next, calculate :

.

Now, find the remainder of 27 when divided by 7:

with a remainder of 6.

Therefore:

.

Thus, the least non-negative remainder when is divided by 7 is: 6