correct answer is: b) 99422

Explanation

We know,
The greatest number of five digits is ‘99999’.
First, we find the LCM of 25,35,45 and 60.
LCM of 25,35,45 and 60 is =5\times3\times5\times7\times3\times4
\rightarrow 6300.
Now, if we divide 99999 by 6300, then we get 15 as quotient and 5499 as remainder.
Now, if we subtract the remainder 5499 from 99999, then we get :
\rightarrow\left(99999-5499\right)=94500.
\therefore 94500 is the number which is divisible by 25,35,45 and 60.
Now, if we add 94500 to 7678, we get =\left(94500+7678\right)
\rightarrow 102178.
But, 102178 is not divisible by the LCM 6300.
So, we have to find the number right after 102178, which is divisible by 6300.
Now, if we check, we find =\left(6300\times17\right)=107100.
\therefore 107100 is the next number after 102178, which is divisible by 6300.
Now, if we find the difference between 107100 and 102178, we get :
\left(107100-102178\right)=4922.
And if we add 4922 to 94500 then we get =\left(94500+4922\right)
\rightarrow 99422.
So that, 99422 is the greatest five digits number whose sum with 7678 is divisible by 25,35,45 and 60.
Ans: 99422 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?