Test For Divisibilty
Test for divisibility are short ways to determine the factors of a natural number without doing the actual calculation.
Divisibility Test for 2:
All even numbers are divisible by 2.
Example: 4, 24, 98, 28890
Divisibility Test for 3:
A number is divisible by 3 if the sum of the digits of the number is divisible by 3. Like 190191 is divisible by 3 because the sum of the digits is 1+9+0+1+9+1=21 and 21 is divisible by 3
Sample problem:
Find the smallest and largest value of a in 781a01 so that the number is divisible by 3.
Solution:
Sum of digits is 7+9+1+a+0+1=a+17. To get the smallest value start testing from 0 but if a=0. 17 is not divisible by 3.Test 1, 18 is divisible by 3. So smallest vale of a=1. To get the largest possible value of a, start counting from 9. After testing the largest value of a=7.
Divisibility Test for 4
A number is divisible by 4 if the the last two digit of the number is divisible by 4. 8934239012344 is divisible by because 44 is divisible by 4.
Sample problem:
For what value/values of a so that the number 89769793048475873a2 is divisible by 4?
Solution:
Just focus your attention to last two digit which is a2. test each number from 0-9.
Possible value of 2 are: {1,3,5,7,9}
Divisibility Test for 5:
A number is divisible by 5 if the last digit of the number is either 0 or 5. 3645348586450 and 883943245 are both divisible by 5.
Sample problem:
For what value(s) of a in 998435a0030 such that the number is divisible by and 5?
Solution:
the last digit of the number is 0. And as mention above any value of a will not affect the property of the number. So possible value of a={1,2,3,4,5,6,7,8,9,0}
Divisibility Test for 6:
the number is divisible by 6 if the number if the number is even and divisible by 3. The number 23142 is divisible by 6 since it is even and the sum of the digits is divisible by 3.
Sample problem:
A number 39499a10 is divisible by 6. What is the smallest value of a?
Solution:
We need to achieve 2 conditions. Even number and divisible by 3. The number is already even so the number must be divisible by 3.
sum of digits= 3+9+4+9+9+a+1+0=35+a. Smallest value of a=1.
Divisibility Test for 7:
To test if the number is divisible by 7. We need to follow some steps. Is 1757 divisible by 7? To answer this question we will do the following steps.
STEPS
- Double the last digit (7*2=14)
- Subtract the product from step 1 and subtract from the original number without the last digit. (175-14=161)
- If the difference is divisible by 7 then the number is divisible by 7. If the number is big repeat step 1 and 2.
Repeating the process it will end up to 16-2=14 which is divisible by 7. Therefore 1757 is divisible by 7.
Divisibility Test for 8:
If the last three digit of the number is divisible by 8 the number is divisible by 8. So we need to perform long division but will only take the last three digit of the dividend. 97974578645968 is divisible by 8 because 968 is a multiple of 8.
Sample problem:
What is the value of a in 8349487512a so that the number is divisible by 8?
Solution:
12a must be divisible by 8. 12a is also 120+a. 120 is multiple of 8. So a=8 is the only solution.
Divisibility Test for 9:
A number is divisible by 9 if the sum of the digits of a number is also divisible by 9918 is divisible by 9 because the sum of digits is (9+9+1+8=27) is divisble by 9.
Sample problem:
For what value of a in 84a09 is the number divisible by 9?
Solution:
The sum of the digits is 8+4+a+0+9=a+21. So a=6 for the digit to be divisible by 9.
Dan
Latest posts by Dan (see all)
- 2016 MMC Schedule - November 4, 2015
- 2014 MTAP reviewer for Grade 3 - September 30, 2015
- 2015 MTAP reviewer for 4th year solution part 1 - August 22, 2015