No Calculators Permitted

You don't need to need to multiply out the numbers. You just need count the powers of 2 and 5.

Divisibility test of $5^n$ is that last $n$ digits of the number must be divisible by $5^n$ . It is clear that 123648 is not divisible by 5 (last digit is 8). So let's just test on 461250.

Through experimenting,

$5|0 \\ 5^2|50 \\ 5^3|250 \\ 5^4|1250 \\ 5^5\nmid 61250$

Hence the largest power of 5 in the product is $5^4$ .

In a similar way, we check if $2^4$ divides either of the numbers or not (remember the divisibility test). It does and hence the number of trailing zeroes are 4. Make sure you understand why we directly checked whether the product is divisible by $2^4$ ( hint: the trailing zeroes cannot be more than 4).

The key to answer this problem is to find powers of 10 that can be formed. Since 10=2×5, we need to find the powers of 5&2. Since 123648 is not divisible by 5 let's chech the powers of 5 in 416250. On factoring the number we can see that there are 4 5s and on factorising 123648 we have more than 4 2s. Since 10 is composed of equal no. Of 2&5, four 10s can be formed. Hence there will be four trailing zeroes.