### Common Errors in Secondary Mathematics

Common Errors Committed  by the  Students  in Secondary Mathematics   Errors  that students often make in doing secondary mathematics  during their practice and during the examinations  and their remedial measures are well explained here stp by step.  Some Common Errors in Mathematics

# DIVISIBILITY TEST INMATHEMATICS

Short cut method of checking the divisibility of numbers. Divisibility test for 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 with examples and explanation

Introduction: Introducing  divisibility tests in classes 6 and 7 can enhance students' understanding, retention, problem-solving skills, and confidence in mathematics. It provides a dynamic and engaging learning experience, fostering a solid foundation for further mathematical concepts.

Divisibility Test :
Divisibility tests are the short-cut methods of checking whether the given number is divisible by the another given digit or number.

Here in this topic we will discuss the divisibility test by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 with examples for more understanding.

Divisibility Test by 2:
If unit digit of a number is even ( 0, 2, 4, 6, 8) then that number is divisible by “2”
All even numbers are divisible by “2”.

Example 1:
Let given number is 2,58,694
Unit digit of this number is “4” it is even. Therefore 258694 is also divisible by “2”

Example 2:
Let given number is 25,86,947
Unit digit of this number is “7”, it is odd. So the number 2586947 is not divisible by “2”.

Divisibility Test by 3:
If sum of the digits of a given number is divisible by “3” then that number is also divisible by “3”.

Example 1:
Let given number is : 3,45,897
Sum of the digits: 3 + 4 + 5 + 8 + 9 + 7 = 36
36 is divisible by “3”, so 345897 is also divisible by “3”

Example 2:
Let given number is :- 9,85,897
Sum of the digits: 9 + 8 + 5 + 8 + 9 + 7 = 46
As 46 is not divisible by “3”, so 985897 is also not divisible by “3”

Divisibility Test by 4:
Take last two digits of of a given number and divide it by “4”, if it is divisible by “4”, then the given number is also divisible by “4”.
If last two digits of a given number are “0, 0” then that number is also divisible by “4”.

Example 1:
Let given number is 37,984
Last two digits : 8, 4,
As  84 is divisible by “4”. So 37984 is also divisible by “4”

Example 2:
Let given number is 8,39,842
Last two digits : 4, 2,
As 42 is not divisible by “4”. So 839842 is also not divisible by “4”

Example 3:
Let given number is 97,39,800
Last two digits : 0, 0 , yes 9739800 is also divisible by “4”

Divisibility Test by 5:
If last digits of a given number is either “0” or “5” then the given number is also divisible by “5”.

Example 1:
Let given numbers are 3,79,520, 7,89,735
Last digit “0” and “5” . Yes above both numbers are divisible by 5

Example 2:
Let given number is 3,79,523
Last digit “3” So above number is not divisible by 5

Divisibility Test by 6:
If a number is divisible by both “2” and “3” then that number is also divisible by 6.

Example 1:
Let given number is 3,57,846
Last digit is “6”, it is even. So it is divisible by “2”

Sum of the digits: 3 + 5 + 7 + 8 + 4 + 6 = 33,
As “33” is divisible by “3”. So 357846 is also divisible by “3”

Here we find that 357846 is divisible by both “2” and “3”. Hence 357846 is also divisible by “6”

Divisibility Test 7:
Algorithm
• Take last digit of a given number.
• Now subtract the double of this digit from the remaining number.
• Now see whether the result is divisible by "7" or not.
• If judgment is still difficult then repeat the above process again and again, till you left with two digit number then make the judgment.

Example 1:
Let given number is 21,875
Last digit “5”. Double of 5 is = 10
Now find 2187-10 = 2,177
Last digit “7”. Double of “7” is = 14
Now find 217 - 14 = 203
Last digit “3”. Double of 3 is = 6
Now find 20-6 = 14
Since 14 is divisible by “7” so 21875 is also divisible by “7”

Example 2:
Let given number is 7,89,423
Last digit “3”. Double of 3 is = 6
Now find 78942 – 6 = 78936
Last digit “6”. Double of 6 is = 12
Now find 7893-12 = 7881
Last digit “1”. Double of 1 is = 2
Now find 788-2 = 786
Last digit “6”. Double of 6 is = 12
Now find 78-12 = 76
76 is not divisible by “7” so number 789423 is not divisible by “7”

Divisibility Test by 8:
If number formed by last three digits of a given number is divisible by “8” then the given number is also divisible by “8”.

If last three digits of a given number are three zero (000) then that number is also divisible by 8.

Example 1:
Let given number is : 57,93,512
Last three digits : 5, 1, 2,
As "512" is divisible by “8” (512/8 = 64) so 5793512 is also divisible by “8”

Example 2:
Let given number is : 5793515
Last three digits : 5, 1, 5,
515 is not divisible by “8”, so 5793515 is also not divisible by “8”

Example 3:
Let given number is : 57,93,000
Last three digits : 0,0,0
So  5793000 is  divisible by “8”

Divisibility Test by 9:
If sum of the digits of a given number is divisible by “9” then that number is also divisible by “9”.

Example 1:
Let given number is : 63,918
Sum of the digits: 6 + 3 + 9 + 1 + 8 = 27
27 is divisible by “9”, so this number is also divisible by 9”

Example 2:
Let given number is : 63,925
Sum of the digits: 6 + 3 + 9 + 2 + 5 = 25
25 is not divisible by “9”, so number 63925 is also not divisible by “9”

Divisibility Test by 10:
If last digits of a given number is “0” then the given number is also divisible by “10”.

Example 1:
Let given number is : 63,910
Last digit is “0”, so 63910 is divisible by “10”

Example 2:
Let given number is : 63,917
Last digit is “7” so 63917 is not divisible by “10”

Divisibility Test by 11:
Mark the digits of a given number odd or even, starts from the right side (unit digit).
Add the digits at odd position.
Add the digits at even places.
Find the difference between the two sums.

Two case arises
1.) If difference is either “0” or divisible by “11” then that number is also divisible by 11.
2.) If the difference is not divisible by 11 then that number is not divisible by 11.

Example 1:
Let given number is : 6,59,175
Sum of the digits at even place = 7 + 9 + 6 = 22
Sum of the digits at odd places = 5 + 1 + 5 = 11
Difference: 22-11 = 11
Since difference is divisible by “11” so the number 659175 is also divisible by “11”

Example 2:
Let given number is : 9,59,171
Sum of the digits at even place = 1 + 1 + 5 = 7
Sum of the digits at odd places = 7 + 9 + 9 = 25
Difference: 25-7= 18
Since difference”18” is not divisible by “11” so the number 959171 is not divisible by “11”