Divisibility by 18

A number that is divisible by 18 must be divisible by 2 and by 9. It also follows that numbers divisible by both 2 and 9 must be divisible by 18.

Therefore, to find out if a number is divisible by 18 or not, we must test for its divisibility by both 2 and 9.

Numbers are divisible by 2 if their last digits are even – 0, 2, 4, 6 or 8.

According to most Math textbooks or reference books, a number is divisible by 9 if the sum of its digits is divisible by 9.

We can see that in our problem for this week all the choices end in even digits, so all must be divisible by 2. We only have to determine which one is exactly divisible by 9.

But instead of finding the sum of the digits of all the choices which may be time consuming considering the length of the numbers, we will use the method of casting out 9s which is discussed thoroughly in the first chapter of our book “Algebra Made Easy as Arithmetic (AMEA)” (Free shipping available until June 5, 2022  if you will purchase out “Olympiad” bundle – AMEA, 25 Math Short Cuts and Inspirational Maths from India at https://www.facebook.com/MATHInicPhils).

Using this method, we can remove or “cast out” any digits adding up to 9 or a multiple of 9. This will shorten the process of adding the digits. In fact, in this example, we can give the answer “By Mere Observation.”

All choices have repeating patterns: a)27…; b) 63…; c) 81…; d) 234… and e) 45… that sum up to 9. Therefore, they all can be cast out. However only in option d can we see the patterns completed. We have five complete sets of the pattern 234. Thus, d is the correct answer.

In all other choices, the number starts and end with the same digits. So, in those options, if we cast out the nines, we will be left with a single digit not equal to 9 and therefore they are not divisible by 9. It would be interesting to note that if we will remove the first digits of those four other choices, they will all be divisible by 18.

Exercises: State whether the number is divisible by 18 or not

1. 91351351351359 →
2. 32342342342346 →
3. 25225225252252 →
4. 40540540540545 →
5. 77220022777722 →
6. 39999999999963 →
7. 56756799567567 →
8. 33366666633363 →
9. 12345678234567 →
10. 43234323432345 →

Answer to Last week’s exercises:

1. 52 ÷ 9 = 5 r. 7
2. 71 ÷ 9 = 7 r. 8
3. 123 ÷ 9 = 13 r. 6
4. 142 ÷ 9 = 15 r. 7
5. 2121 ÷ 9 = 235 r. 6
6. 3113 ÷ 9 = 345 r. 8
7. 11221 ÷ 9 = 1246 r. 7
8. 22111 ÷ 9 = 2456 r. 7
9. 300211 ÷ 9 = 33356 r. 7
10. 321001 ÷ 9 = 35666 r. 7