Latest Elitmus Aptitude Question SOLUTION: How many six digits no can be formed using the digits 1 to 6,without repetation such that the number is divisible by the digit at its unit place?? (a)402 (b)528 (c)648 (d)720

648
when unit digit is 1 =5!
_ _ _ _ _ 2 = 5!
_ _ _ _ _ 3 = 5!
when last two digits are 2 4 and 64 = 4! * 2
_ _ _ _ _ 5 = 5!
_ _ _ _ _ 6 = 5!

total 5*5!+2*4! =648

528
_ _ _ _ _ 2 = 5!
_ _ _ _ _ 3 = 5
_ _ _ _ 2 4 and 64 = 4! * 2
_ _ _ _ _ 5 = 5!
_ _ _ _ _ 6 = 5!

when unit digit is 1=5!
-------------------2=5!
-------------------3=5!
when last two digit's are 24,64 =2*3!
-------------------5=5!
-------------------6=5!
total 5*5!+2*4! =648

Since, there is no repetition, numbers 1 to 6 will always be present in the 6 digit number.

We will go through this step by step.

Divisible by 1 and 1 at the unit place:

_ _ _ _ _ 1

This gives us total of 5 x 4 x 3 x 2 x 1 = 120 numbers.

Divisible by 2 and 2 at the unit place:

_ _ _ _ _ 2

This gives us total of 5 x 4 x 3 x 2 x 1 = 120 numbers.

Divisible by 3 and 3 at the unit place:

_ _ _ _ _ 3

Since, any number will have all the digits from 1 to 6 and the sum 1 + 2 + 3 + 4 + 5 + 6 = 21 is divisible by 3

This gives us total of 5 x 4 x 3 x 2 x 1 = 120 numbers.

Divisible by 4 and 4 at the unit place:

_ _ _ _ _ 4

Here, there are two cases
_ _ _ _ 2 4

This gives 4 x 3 x 2 x 1 = 24 numbers.

_ _ _ _ 6 4
This gives 4 x 3 x 2 x 1 = 24 numbers.
This gives us total of 24 + 24 = 48 numbers.

Divisible by 5 and 5 at the unit place:

_ _ _ _ _ 5

This gives us total of 5 x 4 x 3 x 2 x 1 = 120 numbers.

Divisible by 6 and 6 at the unit place:

_ _ _ _ _ 6

As all 6 digit numbers formed with 1 to 6 digits(without repetition) are divisible by 3 and numbers with 6 at the unit place are even.

This gives us total of 5 x 4 x 3 x 2 x 1 = 120 numbers.

None of these cases will have numbers overlapping with each other.

So, Total numbers = 120 + 120 + 120 + 48 + 120 + 120 = 648

in all the soln u guys have taken 24 and 64 for 4 but why haven't u considered 12,16,16,56,52,32. these are also divisible by 4 .
so the answer will be
_ _ _ _ _ 1 =5!
_ _ _ _ _ 2 =5!
_ _ _ _ _ 3 =5!
_ _ _ _ _ 4=4!
_ _ _ _ _ 5=5!
4!+5!*5=624