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Tests for Divisibility
Divisible by 2n : If last n digits of a number is divisible by 2n, the whole number is divisible by 2n.
Divisible by 2 : A number is divisible by 2, when its unit’s digit is even or 0 (i.e. is divisible by 2 ). e.g. 256, 410, 742, etc.
Divisible by 4 : A number is divisible by 4, when its last two digits are divisible by 4. e.g. 256, 14580, 325732, etc.
Divisible by 8 : A number is divisible by 8, when its last three digits are divisible by 8. e.g. 15256, 57120, 9827832, etc.
Divisible by 16 : A number is divisible by 16, when its last four digits are divisible by 16. e.g. 15248, 57120, 9827824, etc.
Divisible by 3 : A number is divisible by 3, when the sum of its digits is divisible by 3. e.g. 456, 1596, 47613 etc.
Divisible by 9 : A number is divisible by 9, when the sum of its digits is divisible by 9. e.g. 17352, 529056, 513 etc.
Note : (i) The difference between any given number and a number obtaining by permuting the digits of given number is always divisible by 9.
e.g. 53412 – 14352 = 39060, which is divisible by 9.
(ii) The difference between any given number and a number obtaining by adding all the digits of given number is always divisible by 9.
e.g. 53412 – ( 5 + 3 + 4 + 1 + 2 )
= 53412 – 15 = 53397, which is divisible by 9.
Divisible by 5n : If last n digits of a number is divisible by 5n, the whole number is divisible by 5n.
Divisible by 5 : A number is divisible by 5, when its unit’s digit is 5 or 0 (i.e. is divisible by 5 ). e.g. 265, 410, 745, etc.
Divisible by 25 : A number is divisible by 25, when its last two digits are divisible by 25. e.g. 375, 4500, 523250, etc.
Divisible by 125 : A number is divisible by 125, when its last three digits are divisible by 125. e.g. 27625, 58375, 250, etc.
Divisible by 10n : If last n digits of a number is divisible by 10n, the whole number is divisible by 10n.
Divisible by 10 : A number is divisible by 10, when its unit’s digit is 0. (i.e. is divisible by 10 ). e.g. 260, 410, 740, etc.
Divisible by 100 : A number is divisible by 100, when its last two digits are 00 (i.e. is divisible by 100 ). e.g. 300, 4500, 5232000, etc.
Divisible by 1000 : A number is divisible by 1000, when its last three digits are 000 (i.e. is divisible by 1000 ). e.g. 27000, 5837000, 20000, etc.
Divisible by 7 : A number divisible by 7 if and only if the number of tens added to 5 times the number of units is divisible by 7.
e.g. (i) 175 is divisible by 7. Since 17 + 5 5 = 17 + 25 = 42, which is also divisible by 7
(ii) 3423 is divisible by 7. Since 342 + 3 5 = 342 + 15 = 357
and 35 + 7 5 = 35 + 35 = 70. Which is also divisible by 7.
■ Divisible by 11 : A number divisible by 11, when the difference between the sum of the digits in the odd and even places is ‘0’ or a multiple of ‘11’.
e.g. 9473739. Here (9 + 7 + 7 + 9) = 32 and (4 + 3 + 3) = 10
32 – 10 = 22, which is a multiple of 11.
Note : When any number with an even number of digits is added to its reverse, the sum is always a multiple of 11.
e.g. 2341 + 1432 = 3773 which is divisible by 11.
■ Divisible by 13 : A number is divisible by 13 if and only if the number of tens added to four times the number of units is divisible by 13.
e.g. 637 is divisible by 13. Since 63 + 4 7 = 63 + 28 = 91, which is divisible by 13.
■ Divisible by 17 : A number is divisible by 17 if only the number of tens added to twelve times the number of units is divisible by 17.
e.g. 952 is divisible by 17. Since 95 + 12 2 = 95 + 24 = 119, which is divisible by 17.
■ Divisible by 19 : A number is divisible by 19 if and only if the number of tens added to twice the number of units is divisible by 19.
e.g. (i) 76 is divisible by 19 since 7 + 2 6 = 7 + 12 = 19 which is divisible by 19.
If a number is divisible by ‘x’ and ‘y’, then it is divisible by the [x, y].
( Where, [x, y] = L.C.M. of x and y. )
Divisible by 6 : A number is divisible by 6 if it is divisible by both 2 and 3.
e.g. 354
Divisible by 12 : A number is divisible by 12 if it is divisible by both 3 and 4.
e.g. 624
Divisible by 14 : A number is divisible by 14 if it is divisible by both 7 and 2.
e.g. 798
Divisible by 15 : A number is divisible by 15 if it is divisible by both 3 and 5.
e.g. 645
Divisible by 18 : A number is divisible by 18 if it is divisible by both 9 and 2.
e.g. 882
If a number is divisible by ‘p’ as well as by ‘q’, where ‘p’ and ‘q’ are co-primes ( or relatively prime ) , then the given number is divisible by ‘pq’.
[ Where, co-prime means ( p, q ) = 1 ]
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’.
Examples :
i. Show that 183572 is divisible by both 2 and 4, but not divisible by 8. Why so?
Solution. Given number ends in 2. so, it is divisible by 2.
Number formed by last 2 digits of the given number is 72, which is divisible by 4. So, the given number is divisible by 4.
But, the number formed by the last 3 digits of the given number is 572, which is not divisible by 8.
So, the given number is not divisible by 8.
Clearly, 2 and 4 are not co-primes. So, the given number need not be divisible by their product, namely 8.
( and also we know that, if a number is divisible by ‘a’ and ‘b’ then it is divisible by [a, b]. here, [2, 4] = 4 8.)
ii. Show that 3953928 is divisible by 88.
Solution. The number formed by the last 3 digits of the given number is 928, which is divisible by 8.
So, the given number is divisible by 8.
In the given number, we have :
(Sum of digits at odd places) – (Sum of digits at even places)
= [(3 + 5 + 9 + 8) – (9 + 3 + 2)] = 11, which is divisible by 11.
So, the given number is divisible by 11.
Thus, the given number is divisible by both 8 and 11, where 8 and 11 are co-primes.
Hence, the given number is divisible by 88.
( or the number is divisible by [8, 11] = 88 )
