Problems based on the divisibility test

1. Replace the sign ___ by the smallest number to make them divisible by the given number.

a. 7158__ by 6

By the divisibility test,
if the number is divisible by 2 and 3 then that number is divisible by 6.
if the given number is divisible by 2 the units place must be 0 , 2, 4 , 6 , or 8.
if the given number is divisible by 3 then first we have to add the digits 7+1+5+8=21
21 is divisible by 3
so we can replace the sign by 0 since 71580(unit place 0) which is divisible by 2 and 71580(add will get 21) which is divisible by 3.
hence 71580 is divisible by 6

b. 8152__ by 4

By the divisibility test,
if the last two digits is divisible by 4 then that number is divisible by 4.
if we replace the sign by 0 then the last two digits be 20 which is divisible by 4.
hence 81520 is divisible by 4

c. 86__72 by 11

By the divisibility test,
the sum of the numbers in odd place = 8 + __+2  = 10+__
sum of the numbers in even place = 6 + 7  = 13
10+__  - 13 = 0 (since the difference must be either 0 or divisible by 11)
[we have to write the smallest possible number hence we choose 0]
so __ be 3
13-13=0
hence 86372 is divisible by 11

d. 631__24 by 8

By the divisibility test,
if the last three digit is divisible by 8 then that number is divisible by 8.
__24 , 24 itself is divisible by 8 hence we can replace the sign by 0
hence 631024 is divisible by 8.

2. State whether the following sentences are true or false.

a. 231648 is divisible by 4 and 8

Divisible by 4
lets consider last two digits 48 which is divisible by 4.
Divisible by 8
lets consider last three digits 648 which is divisible by 8
hence 231648 is divisible by 4 and 8 TRUE

b. 976575 is divisible by 3 and 15

Divisible by 3
lets add all the digits 9+7+6+5+7+5=39 which is divisible by 3
Divisible by 15
if the number is divisible by 3 and 5 then that number is divisible by 15
we already found that number number is divisible by 3
the unit place is 5 so the given number is divisible by 5 
hence it is divisible by 15
976575 is divisible by 3 and 15 TRUE


c. 567835 is divisible by 5 and 25  

Divisible by 5
the unit place in the given number 5, so the given number is divisible by 5
Divisible by 25
if the last two digits are 00,50 or 75 then that number is divisible by 25
here the last two digits is 35, so 567835 is not divisible by 25
567835 is divisible by 5 and 25 FALSE  

3. What digit you add each time to make the numbers on the Violet divisible by the numbers given in the RED

a. 67984311 by 2,3,4,5,6,8,9,10

Divisible by 2
if we add last digit as 0 then according to the divisibility test,
679843110 is divisible by 2

Divisible by 3
lets add all the digits 6+7+9+8+4+3+1+1+__ = 39+__
39 itself is divisible by 3
hence we can replace the sign as 0
679843110 is divisible by 3

Divisible by 4
the last two digits 1__, if we replace sign by 2 then 12 which is divisible by 4
hence 679843112 is divisible by 4

Divisible by 5
the last digit must be either 0 or 5
hence it can be 679843110 or 679843115 which are divisible by 5

Divisible by 6
According to the divisibility test,
the number must be divisible by both 2 and 3
if we add 0 that is 679843110 is divisible by both 2 and 3.
hence 679843110 is divisible by 6

Divisible by 8
lets consider the last three digits 11__ ,
110,111 are not divisible by 8
112 is divisible by 8
hence 679843112 is divisible by 8

Divisible by 9
Sum of the digits = 39 + __ 
39 is not divisible by 9
if we add 6 to 39 which gives 45 and 45 be divisible by 9
hence 679843116 is divisible by 9

Divisible by 10
the last digit must be 0
hence 679843110 is divisible by 10