Understanding Number Systems - 2 - Divisibility of Numbers.

This is the 2nd post on Number Systems. For 1st post, click here.

Todays post will deal with Divisibility rules of Numbers. This is very useful when finding if a number is Prime or not, factoring a number etc.

Rules:

1. A number is said to be divisible by 2 if its unit digit is 0 or even. Eg (14, 998, 2000 etc)
2. A number is said to be divisible by 3 if the sum of its digits is divisible by 3. (Eg : 1431, 6, 27 etc).
3. A number is said to be divisible by 4 if its last two digits are 0 each or the number formed by them is completely divisible by 4. (Eg: 100, 9999904 etc)
4. A number is said to be divisible by 5 if its last digit is 0 or 5.
5. A number is said to be divisible by 6 if its unit digit is 0 or even and sum of its digits is divisible by 3. (6, 18, 96 etc).
6. A number is said to be divisible by 8 if its last three digits are 0 each or the number formed by them is completely divisible by 8 (Eg: 16, 128, 1111128 etc).
7. A number is said to be divisible by 9 if the sum of its digits is divisible by 9. (Eg 9, 9801, 774 etc).
8. A number is said to be divisible by 10 if its unit digit is 0. (Eg 10, 100 etc)
9. A number is said to be divisible by 11 if the difference of the sum of its digits in the odd place and the sum of its digits in the even place is either 0 or a multiple of 11 ( Eg. 121, 1331 etc)

Divisibility by 11 using examples:

Consider the number 187968.

Since the difference 11 is divisible by 11,  187968 is divisible by 11!



Now consider the number 1754.

Since the difference 5 is not divisible by 11,  1754 is not divisible by 11!
Also, the reminder when you divide a number by will be the reminder when this difference is divided by 11. So, in this case, 1754 will leave a reminder 5 when divided by 11!

Reminder Theorems

Above, we have seen rules to check if a number is divisible by another number. But often in exams, getting to know the reminder quickly is also very important (Questions like what number should be added to X to make it divisible by a,b,c).

Reminder of Number when Divided by 3  = Reminder of sum of digits of a number when divided by 3. Eg 1112 will leave reminder 2 when divided by 3. (1+1+1+2 % 3 = 5%3 = 2)

Reminder of Number when Divided by 4  = Reminder of last 2 digits of a number when divided by 4. Eg: 1233471 will leave reminder 3 when divided by 4 since 71 (last 2 digits) leaves reminder 3 when divided by 4.

Reminder of Number when Divided by 5  = Reminder of last digit of a number when divided by 5. Eg: 1233471 will leave reminder 1 when divided by 5 since 1 (last digit) leaves reminder 1 when divided by 5.

Excercise Questions 

1. Is the number 881 prime? [Easy]

2. Factorize 187968 [Medium]

3. Whats the smallest number should be added to 156789 to make it divisible by 11? [Medium]

4. Whats the smallest number that should be added to 677 to make it divisible by 4, 5, and 11? [Hard]

Answers to Previous Excercise:

1. 7691 - (58+374+1693+2085) = 3481

2.  a.  What happens when you multiply 3 even numbers?  - Even Number

     b.  2 even numbers and an odd number? - Even Number

     c.  2 odd numbers and an even number? - Even Number

     d.  and 3 odd numbers? - Odd Number.

3. a. What happens when you multiply 'N' even numbers?  - Even Number
    b. and 'N' odd numbers?- Odd Number

4. Ram multiplied 4 consecutive natural numbers and said the product was odd. Is Ram telling the truth?- No, Product of 2 or more consecutive natural numbers is always even!

Next Post will talk about Powers of Numbers, reminders when a^b is divided by some number c etc.