Page 68 - ICSE Math 8

Test of Divisibility by 6
A number is divisible by 6 if it is divisible by both 2 and 3.
For a number in the generalized form:
(a) A 2-digit number 10a + b is divisible by 6 if b is 0, 2, 4, 6 or 8 and a + b is divisible by 3.
(b) A 3-digit number 100a + 10b + c is divisible by 6 if c is 0, 2, 4, 6 or 8 and a + b + c is divisible by 3.
For example, 18, 48, 132, 324, 408, etc., are divisible by 6.
Test of Divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9.
For a number in the generalized form:
(a) A 2-digit number 10a + b is divisible by 9 if a + b is divisible by 9.
(b) A 3-digit number 100a + 10b + c is divisible by 9 if a + b + c is divisible by 9.
For example, 63, 108, 315, 432, 738, etc., are divisible by 9.
Test of Divisibility by 10
A number is divisible by 10 if its digit at the ones place is 0.
For a number in the generalized form:
(a) A 2-digit number 10a + b is divisible by 10 if b = 0.
(b) A 3-digit number 100a + 10b + c is divisible by 10 if c = 0.
For example, 40, 90, 270, 480, 950, etc., are divisible by 10.
Test of Divisibility by 11
A number is divisible by 11 if the difference between the sum of its digits at odd places and sum of its digits
at even places is either 0 or a multiple of 11.
For a number in the generalized form:
A 3-digit number 100a + 10b + c is divisible by 11 if (a + c) – b = 0 or (a + c) – b is a multiple of 11.
For example, 66, 99, 264, 341, 495, etc., are divisible by 11.
Example 4: If 21y5 is a multiple of 9, where y is a digit, then what is the value of y?
Solution: 21y5 is a multiple of 9.
fi 2 + 1 + y + 5 = 8 + y should also be a multiple of 9.
fi 8 + y = 9 or 18 or 27 or 36 and so on.
8 + y = 9 fi y = 1 ; 8 + y = 18 fi y = 10 ; 8 + y = 27 fi y = 19 and so on.
Since y is a digit, so y = 1.

Example 5: Given that the number 1372x413 is divisible by 11, where x is a digit, find the value of x.
Solution: Given number = 1372x413
Sum of digits at odd places = 1 + 7 + x + 1 = 9 + x
Sum of digits at even places = 3 + 2 + 4 + 3 = 12
Difference = (x + 9) – 12 = x – 3
Given, 1372x413 is divisible by 11.
fi x – 3 is divisible by 11.
fi x – 3 = 0 or 11 or 22 or 33 and so on.
x – 3 = 0 fi x = 3 ; x – 3 = 11 fi x = 14
x – 3 = 22 fi x = 25 and so on.
Since x is a digit, so x = 3.

EXERCISE 5.2

1. Give three numbers which are divisible by 2 but not by 4. Is there any number which is divisible by 4
but not by 2?
2. Give two examples of a number which is divisible by: (a) 3 but not 9 (b) 2 but not 6 (c) 2 and 4 but not 8

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