Problem Solving

talaangoshtari wrote:How many five digit positive integers that are divisible by 3 can be formed using the digits 0, 1, 2, 3, 4 and 5, without any of the digits getting repeating

A. 15
B. 96
C. 216
D. 120
E. 625

If a number is divisible by 3, the SUM of the digits will be divisible by 3.
0 + 1 + 2 + 3 + 4 + 5 = 15 (which is divisible by 3)

We need to remove 1 digit (to create a 5-digit number). So, to ensure that the SUM of the remaining 5 digits is divisible by 3, the digit that we remove must be divisible by 3.
That means, we can remove EITHER 0 or 3

Removing 0 leaves us with the digits 1, 2, 3, 4, and 5, which have a sum of 15. Great.
Removing 3 leaves us with the digits 0, 1, 2, 4, and 5, which have a sum of 12. Great.

So, how many 5-digit numbers can we create with the digits 1, 2, 3, 4, and 5, and how many 5-digit numbers can we create with the digits 0, 1, 2, 4, and 5?

Start with the digits 1, 2, 3, 4, and 5
We cannot repeat digits.
So, we have 5 options for the first digit in the number.
We have 4 options for the second digit in the number.
We have 3 options for the third digit in the number.
We have 2 options for the fourth digit in the number.
We have 1 option for the fifth digit in the number.
So, the TOTAL number of 5-digit numbers = (5)(4)(3)(2)(1) = 120

NOTE: we haven't yet counted all of the 5-digit numbers can we create with the digits 0, 1, 2, 4, and 5
This means our final answer must be GREATER than 120.
So, we can ELIMINATE answer choices A, B, and D

IMPORTANT: IF we were to start listing 5-digit numbers that can be created with the digits 0, 1, 2, 4, and 5, we would have to ensure that the first digit is NOT 0. Otherwise, we'd get a 4-digit number (e.g., 02451 is NOT a 5-digit number).
This means that our list of 5-digit numbers (using 0, 1, 2, 4, and 5) will have FEWER THAN 120 numbers.

This means we can ELIMINATE answer choice E.

This leaves only answer choice C

Cheers,
Brent