correct answer is: b) KaTeX parse error: KaTeX doesn't work in quirks mode.

Explanation

We know,
The greatest number of five digits is KaTeX parse error: KaTeX doesn't work in quirks mode.
First, we find the LCM of KaTeX parse error: KaTeX doesn't work in quirks mode. and KaTeX parse error: KaTeX doesn't work in quirks mode.
LCM of KaTeX parse error: KaTeX doesn't work in quirks mode. and KaTeX parse error: KaTeX doesn't work in quirks mode. is KaTeX parse error: KaTeX doesn't work in quirks mode.
KaTeX parse error: KaTeX doesn't work in quirks mode.
Now, if we divide KaTeX parse error: KaTeX doesn't work in quirks mode. by KaTeX parse error: KaTeX doesn't work in quirks mode. then we get KaTeX parse error: KaTeX doesn't work in quirks mode. as quotient and KaTeX parse error: KaTeX doesn't work in quirks mode. as remainder.
Now, if we subtract the remainder KaTeX parse error: KaTeX doesn't work in quirks mode. from KaTeX parse error: KaTeX doesn't work in quirks mode. then we get KaTeX parse error: KaTeX doesn't work in quirks mode.
KaTeX parse error: KaTeX doesn't work in quirks mode.
KaTeX parse error: KaTeX doesn't work in quirks mode. is the number which is divisible by KaTeX parse error: KaTeX doesn't work in quirks mode. and KaTeX parse error: KaTeX doesn't work in quirks mode.
Now, if we add KaTeX parse error: KaTeX doesn't work in quirks mode. to KaTeX parse error: KaTeX doesn't work in quirks mode. we get KaTeX parse error: KaTeX doesn't work in quirks mode.
KaTeX parse error: KaTeX doesn't work in quirks mode.
But, KaTeX parse error: KaTeX doesn't work in quirks mode. is not divisible by the LCM KaTeX parse error: KaTeX doesn't work in quirks mode.
So, we have to find the number right after KaTeX parse error: KaTeX doesn't work in quirks mode. which is divisible by KaTeX parse error: KaTeX doesn't work in quirks mode.
Now, if we check, we find KaTeX parse error: KaTeX doesn't work in quirks mode.
KaTeX parse error: KaTeX doesn't work in quirks mode. is the next number after KaTeX parse error: KaTeX doesn't work in quirks mode. which is divisible by KaTeX parse error: KaTeX doesn't work in quirks mode.
Now, if we find the difference between KaTeX parse error: KaTeX doesn't work in quirks mode. and KaTeX parse error: KaTeX doesn't work in quirks mode. we get KaTeX parse error: KaTeX doesn't work in quirks mode.
KaTeX parse error: KaTeX doesn't work in quirks mode.
And if we add KaTeX parse error: KaTeX doesn't work in quirks mode. to KaTeX parse error: KaTeX doesn't work in quirks mode. then we get KaTeX parse error: KaTeX doesn't work in quirks mode.
KaTeX parse error: KaTeX doesn't work in quirks mode.
So that, KaTeX parse error: KaTeX doesn't work in quirks mode. is the greatest five digits number whose sum with KaTeX parse error: KaTeX doesn't work in quirks mode. is divisible by KaTeX parse error: KaTeX doesn't work in quirks mode. and KaTeX parse error: KaTeX doesn't work in quirks mode.
Ans: KaTeX parse error: KaTeX doesn't work in quirks mode. is the required greatest number of five digits.

There’s another way to pose this same math question:

1. Identify the largest natural number with five digits such that the sum with 7678 is divisible by 25, 35, 45 and 60.
2. What is the maximum five-digit integer for which the sum with 7678 is divisible by 25, 35, 45 and 60?