A five-digit number is formed by using each of the digits 1, 2, 3, 4, and 5 exactly once. How many five-digit numbers formed this way are divisible by 12?

Question

A hint was given. Use the fact that for a given number to be divisible by 3, it suffices for the sum of its digits to be divisible by 3.

Answer 1

to be divisible by 12, the number must be divisible by both 3 and 4

how many 5 digit numbers are there ?

5*4*3*2*1=120 5 digit numbers

any 5 digit number formed using 1, 2, 3, 4, and 5 is divisible by 3 because 5+4+3+2+1=15 and 15 is divisible by 3

to be divisible by 4, the two right-most digits of the 5 digit number must form a number divisible by 4

the two right-most digits of each 5 digit number must be either 52, 32, 24, or 12

how many of each are there ?

start with 52

this leaves 1, 3, and 4 to precede 52

how many 3-digit numbers can be formed with 1, 3, and 4 ?

3*2*1=6

they are:

134, 143, 341, 314, 413, 431

the 6 5 digit numbers are:

13452, 14352, 34152, 31452, 41352, 43152

there are 6 numbers that end in 32

there are 6 numbers that end in 24

there are 6 numbers that end in 12

there are 24 5 digit numbers out of the 120 5 digit numbers that are divisible by 12

for the numbers ending in 12, for example, you have:

34512, 35412, 45312, 43512, 54312, 53412