Numbers-II
V. TESTS OF DIVISIBILITY
1. Divisibility By 2:
A number is divisible
by 2, if its unit's digit is any of 0, 2, 4, 6, 8. Example. 84932 is
divisible by 2, while 65935 is not.
2. Divisibility By 3:
A number is divisible
by 3, if the sum of its digits is divisible by 3. Example.592482
is divisible by 3, since sum of its digits = (5 + 9 + 2 + 4 + 8 + 2) = 30,
which is divisible by 3. But, 864329
is not divisible by 3, since sum of its digits =(8 + 6 + 4 + 3 + 2 + 9) = 32,
which is not divisible by 3.
3. Divisibility By 4:
A number is divisible
by 4, if the number formed by the last two digits is divisible by 4. Example. 892648 is
divisible by 4, since the number formed by the last two digits is 48, which is divisible by 4. But, 749282
is not divisible by 4, since the number formed by the last two digits is 82,
which is not divisible by 4.
4. Divisibility By 5:
A number is divisible
by 5, if its unit's digit is either 0 or 5. Thus, 20820 and 50345 are divisible
by 5, while 30934 and 40946 are not.
5. Divisibility By 6:
A number is divisible
by 6, if it is divisible by both 2 and 3. Example. The number 35256 is clearly
divisible by 2. Sum of its
digits = (3 + 5 + 2 + 5 + 6) = 21, which is divisible by 3. Thus, 35256 is
divisible by 2 as well as 3. Hence, 35256 is divisible by 6.
6. Divisibility By 8:
A number is
divisible by 8, if the number formed by the last three digits
of the given number is divisible by 8. Example. 953360 is
divisible by 8, since the number formed by last three digits is 360, which is
divisible by 8. But, 529418
is not divisible by 8, since the number formed by last three digits is 418,
which is not divisible by 8.
7. Divisibility By 9:
A number is
divisible by 9, if the sum of its digits is divisible by 9. Example. 60732 is
divisible by 9, since sum of digits (6 + 0 + 7 + 3 + 2) = 18, which is
divisible by 9. But, 68956 is
not divisible by 9, since sum of digits = (6 + 8 + 9 + 5 + 6) = 34, which is
not divisible by 9.
8. Divisibility By 10 :
A number is
divisible by 10, if it ends with 0. Example. 96410 ,
10480
9. Divisibility By 11:
A number is
divisible by 11, if the difference of the sum of its digits at odd places and
the sum of its digits at even places, is either 0 or a number divisible by 11. Example.
The number 4832718 is divisible by 11, since : (sum of
digits at odd places) - (sum of digits at even places) (8 + 7 + 3 +
4) - (1 + 2 + 8) = 11, which is divisible by 11.
10. Divisibility By 12:
A number is divisible by
12, if it is divisible by both 4 and 3.
Example: Consider the number 34632.
(i) The
number formed by last two digits is 32, which is divisible by 4,
(ii) Sum of
digits = (3 + 4 + 6 + 3 + 2) = 18, which is divisible by 3. Thus, 34632 is
divisible by 4 as well as 3. Hence, 34632 is divisible by 12.
11. Divisibility By 14 :
A number is divisible
by 14, if it is divisible by 2 as well as 7.
12. Divisibility By 15 :
A number is divisible by
15, if it is divisible by both 3 and 5.
13. Divisibility By 16 :
A number is divisible by
16, if the number formed by the last 4
digits is divisible by 16. Example.7957536
is divisible by 16, since the number formed by the last four digits is 7536,
which is divisible by 16.
14. Divisibility By 24 :
A given number is
divisible by 24, if it is divisible by both 3
and 8.
15. Divisibility By 40 :
A given number is
divisible by 40, if it is divisible by both 5 and 8.
16. Divisibility By 80 :
A given number is
divisible by 80, if it is divisible by both 5 and 16.
Note :
If a number is divisible by p as well as q, where p and q are co-primes, then
the given number is divisible by pq. If p and q
are not co-primes, then the given number need not be divisible by pq, even when it
is divisible by both p and q. Example. 36
is divisible by both 4 and 6, but it is not divisible by (4x6) = 24, since 4 and 6 are not co-primes.
