Number System

Step 1: Solve for the number of solutions.
We find the total number of positive integral solutions by factoring the number 30 and finding the divisors. After calculating, we find that the number of solutions is 27.
Thus, the correct answer is (2).

Step 1: Solve the equation.
Given , we solve the equation and find that the possible pairs are 6.
Thus, the correct answer is (2).

Step 1: (3+√(5))²n + (3-√(5))²n is an integer. Step 2: Since 0<3-√(5)<1, 0<(3-√(5))²n<1 Step 3: Hence, the integer just greater than (3+√(5))²n is: (3+√(5))²n+(3-√(5))²n Step 4: This integer is divisible by 2ⁿ-1. Therefore, option (A) is correct.

We need to find how many integers in the range make the expression divisible by 5.

To determine when a number is divisible by 5, we find when . This simplifies our task to checking for what values of modulo 5 this equation holds.

Let's examine the values of :

• If , then , , meaning modulo 5.
• If , then , , meaning modulo 5.
• If , then , , meaning modulo 5.
• If , then , , meaning modulo 5.
• If , then , , meaning modulo 5.

Thus, is divisible by 5 when or .

Counting numbers from 1 to 100:

• Numbers satisfying are .
• Numbers satisfying are .

Both sequences form arithmetic progressions:

• For , the first term is 3, last term is 98, and common difference is 5. The number of terms is .
• For , the first term is 4, last term is 99, and common difference is 5. The number of terms is .

Adding the number of solutions gives numbers between 1 and 100 for which is divisible by 5.

Therefore, there are 20 such integers for each case, and the correct answer is simply the list of integers that results from the overlap of counting 3 and 4 as the same. Hence, the number of distinct integers from to for which the expression is divisible by is 20.