Brishti went on an 8-hour trip in a car. Before the trip, the car had travelled a total of x km till then, where x is a whole number and is palindromic, i.e., x remains unchanged when its digits are reversed. At the end of the trip, the car had traveled a total of $26862 \mathrm{~km}$till then, this number again being palindromic. If Brishti never drove at more than $110 \mathrm{~km} / \mathrm{hr} , then the greatest possible average speed at which she drove during the trip, in km/hr, was
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Official Correct Answer: D. Brishti went on an 8-hour trip in a car. Before the trip, the car had traveled a total of x km, which is a whole number and a palindrome. At the end of the trip, the total distance was 26862 km, which is also a palindrome. She never drove more than 110 km/hr. We find the greatest possible average speed she could have driven during the trip. The options are 80, 90, 110, or 100 km/hr.
Okay, let's break this down. First, palindromic numbers are numbers that read the same forwards and backwards. So, x is a palindrome, and 26862 is also a palindrome. The trip took 8 hours, so the distance she drove during the trip is 26862 - x. Since she didn't drive more than 110 km/hr, the average speed can't exceed 110 km/hr. But we need the greatest possible average speed, so we want to maximize (26862 - x)/8, which means we need to minimize x.
So, the key here is to find the largest possible x that is a palindrome and less than 26862. Because if x is as large as possible, then the distance driven during the trip will be as small as possible, but wait. Wait, no, if x is as large as possible, then the distance driven is 26862 - x, which would be as small as possible, but we want the average speed to be as large as possible. So actually, we need to minimize x to maximize the distance driven, which would then allow for a higher average speed.
Wait, hold on, that seems contradictory. Let me think again. The average speed is (total distance driven)/8. So, to maximize the average speed, we need to maximize the distance driven, which is 26862 - x. Therefore, to maximize 26862 - x, we need to minimize x. So, yes, we need the smallest possible x that is a palindrome less than 26862.
But wait, x is the total distance before the trip, so it must be less than 26862. So, the smallest possible x is the largest palindrome less than 26862. Wait, no, no, wait. If x is the total distance before the trip, then x must be less than 26862. So, to maximize the distance driven, we need the largest possible x, but x has to be a palindrome. Wait, no, wait. If x is as large as possible, then 26862 - x is as small as possible, which would give a lower average speed. So, to get the maximum average speed, we need x to be as small as possible, so that 26862 - x is as large as possible.
But x is a palindrome. So, we need the smallest palindrome less than 26862. But wait, palindromic numbers can be of varying lengths. 26862 is a 5-digit palindrome. So, the smallest palindrome less than 26862 would be... Hmm, let's think.
Wait, 26862 is a palindrome. The next smaller palindrome would be 26762, because if we decrease the middle digit by 1, keeping the rest the same. Let me check: 26862 minus 100 is 26762, which is a palindrome. So, 26762 is a palindrome. Is that the next one? Or is there a smaller one?
Wait, palindromes can also have fewer digits. For example, 26862 is a 5-digit number. The next smaller palindrome could be a 4-digit number, like 2662, but that's way smaller. Wait, but 2662 is a 4-digit palindrome, but 26862 is a 5-digit number. So, the smallest x is not necessarily a 4-digit number; it could be a 5-digit palindrome less than 26862.
Wait, but 26862 is the total distance after the trip. So, x must be less than 26862. So, the largest possible x is 26862 minus the smallest possible distance driven, but we need x to be a palindrome.
Wait, perhaps I'm overcomplicating. Let's approach this systematically.
First, let's note that 26862 is a palindrome. So, x must be a palindrome less than 26862. To maximize the average speed, we need to maximize the distance driven, which is 26862 - x. Therefore, we need to minimize x. So, the smallest possible x is the smallest palindrome less than 26862.
But wait, x is the total distance before the trip, so it's a whole number, and it's a palindrome. So, the smallest possible x is 0, but that's not practical because she went on an 8-hour trip, so she must have driven some distance. But the problem doesn't specify that x is positive, just that it's a whole number and a palindrome. So, technically, x could be 0, but that would make the distance driven 26862 km, which would require an average speed of 26862 / 8 = 3357.75 km/hr, which is way more than 110 km/hr. But the problem states that she never drove at more than 110 km/hr. So, the maximum distance she could have driven is 110 * 8 = 880 km. Therefore, 26862 - x ≤ 880, so x ≥ 26862 - 880 = 25982.
So, x must be a palindrome greater than or equal to 25982 and less than 26862. So, we need to find the largest palindrome less than or equal to 26862, but actually, x must be at least 25982 because 26862 - x ≤ 880.
Wait, so x must be ≥ 25982 and ≤ 26862, and x is a palindrome. So, we need to find the largest palindrome less than or equal to 26862, but also greater than or equal to 25982.
So, let's find the largest palindrome less than or equal to 26862. Since 26862 is a