Final answer:
The smallest value of x that makes the number 65x6 divisible by 11 is 5, as per the divisibility rule of 11. This is because (6 + 5) - (5 + 6) = 5 - 5 = 0, which meets the criteria for divisibility by 11. The correct option is D.
Step-by-step explanation:
To find the smallest value of x such that the number 65×6 is divisible by 11, we need to apply the divisibility rule for 11. The rule states that a number is divisible by 11 if the difference between the sum of the digits in the odd positions and the sum of the digits in the even positions is either 0 or divisible by 11.
Let's denote the number in question as 65x6, where x is the digit we need to find. Sum of digits in odd positions: 6 + x. Sum of digits in even positions: 5 + 6. We calculate the difference: (6 + x) - (5 + 6) = x - 5.
For the number to be divisible by 11, x - 5 must be a multiple of 11 or 0. Testing the options given a) 2 b) 3 c) 4 d) 5, we find that when x = 5, x - 5 = 0, which satisfies the rule. Therefore, the smallest value of x is 5.