Page 56 - ICSE Math 6

Divisibility Tests

• Divisibility by 2: A number is divisible by 2, if its ones place digit is either zero or an even number.
For example, 8, 14, 64, 120, etc., are divisible by 2.
• Divisibility by 3: A number is divisible by 3, if the sum of all of its digits is divisible by 3.
For example, 3,321 is divisible by 3 as, 3 + 3 + 2 + 1 = 9 is divisible by 3.

• Divisibility by 4: A number is divisible by 4, if the number formed by its last two digits (i.e., ones
and tens) is divisible by 4. For example, 25,396 is divisible by 4 as 96 is divisible by 4.
• Divisibility by 5: A number is divisible by 5, if its ones place digit is either 0 or 5. For example,
50, 65, 120, 1,145, etc., are divisible by 5.
• Divisibility by 6: A number is divisible by 6, if the number is divisible by both 2 and 3.
For example, 21,390 is divisible by 2, as its ones place digit is zero. Also, 21,390 is divisible by
3 as, 2 + 1 + 3 + 9 + 0 = 15 is divisible by 3. Therefore, 21,390 is divisible by 6.
• Divisibility by 8: A number is divisible by 8, if the number formed by its last three digits (i.e., ones,
tens and hundreds) is divisible by 8. For example, 45,640 is divisible by 8 as 640 is divisible by 8.

• Divisibility by 9: A number is divisible by 9, if the sum of all of its digits is divisible by 9.
For example, 4,509 is divisible by 9 as, 4 + 5 + 0 + 9 = 18 is divisible by 9.

• Divisibility by 10: A number is divisible by 10, if its ones place digit is 0. For example, 20, 40,
190, 5,330, etc., are divisible by 10.
• Divisibility by 11: A number is divisible by 11, if the difference between the sum of the digits at
its odd places (starting from right) and the sum of the digits at its even places is either zero or a
multiple of 11. For example, 50,875 is divisible by 11 as,
Sum of the digits at odd places = 5 + 8 + 5 = 18
Sum of the digits at even places = 7 + 0 = 7
Difference between the sums = 18 – 7 = 11

Example 10: Consider the numbers 29, 30, 65, 2,198 and 16,791.
(a) Which of these are divisible by 2? (b) Which of these are divisible by 3?
(c) Which of these are divisible by 5? (d) Which of these are divisible by 10?
Solution: (a) In 30 and 2,198, the digits at ones place are 0 and 8 respectively. Thus, 30 and 2,198
are divisible by 2.
(b) Sum of digits of 29 = 2 + 9 = 11, which is not divisible by 3.
Sum of digits of 30 = 3 + 0 = 3, which is divisible by 3.
Sum of digits of 65 = 6 + 5 = 11, which is not divisible by 3.
Sum of digits of 2,198 = 2 + 1 + 9 + 8 = 20, which is not divisible by 3.
Sum of digits of 16,791 = 1 + 6 + 7 + 9 + 1 = 24, which is divisible by 3.
Thus, 30 and 16,791 are divisible by 3.
(c) In 30 and 65, the digits at ones place are 0 and 5 respectively. Thus, 30 and 65 are
divisible by 5.
(d) The only number having 0 at its ones place is 30. So, 30 is divisible by 10.
Example 11: Write a digit in the blank space of each of the following numbers so that the number

formed is divisible by 11.
(a) 7,__4,891 (b) 4,97,6__3

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