Let p, be two-digit numbers neither of which are divisible by 10. Let be the four-digit number by putting the digits of followed by the digits of (in order). As p, vary, computer prints on the screen if gcd(p, q) = 1 and + divides r. Suppose that the largest number that is printed by the computer is N. Determine the number formed by the last two digits of N (in the same order).
Hint 1: Start by analyzing the initial conditions and setting up the basic equations. r = 100r + q.
Hint 2: Look for algebraic properties, symmetry, or geometric theorems to simplify. r + | = 100 + q.
Hint 3: Proceed with the final algebraic steps to solve the system. => + | 99r.
Step 1: r = 100r + q
Step 2: r + | = 100 + q
Step 3: => + | 99r
Step 4: But gcd(r + q, r) = 1
Step 5: => + | 99
Step 6: => + = 33 or 99
Step 7: For N to be maximum + = 99
Step 8: Where = 89, = 13
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