Find the two-digit numbers A and B such that the number 65AB0 is divisible by…

Step-by-Step Solution

1. Analyze the Divisor: To be divisible by 66, a number must be divisible by its co-prime factors, which are 2, 3, and 11.

2. Apply Divisibility Rules:

   • Divisibility by 2: The number 65AB0 ends in 0, so it is even and satisfies the rule for 2.

   • Divisibility by 3: The sum of the digits must be divisible by 3.

   • Divisibility by 11: The alternating sum of the digits must be divisible by 11.

3. Test Option C (A=17, B=17): The number becomes 6517170.

   • Sum of digits: 6 + 5 + 1 + 7 + 1 + 7 + 0 = 27. Since 27 is divisible by 3, the number satisfies the divisibility rule for 3.

   • Alternating sum: (6 + 1 + 1 + 0) - (5 + 7 + 7) = 8 - 19 = -11. Since -11 is divisible by 11, the number satisfies the divisibility rule for 11.