लेखक
ॐ जितेन्द्र सिंह तोमर
M.A., B.Ed., DNYS, MASSCOM
10/7/28/11/2021
05.03 वैदिक गणित (03) || भाग || 3 व 9 की विभाज्यता के नियम
Division:
DIVISIBILITY BY 3
(i) By Digital Sum Method
All numbers with a digit sum of 3, 6 or 9 are always divisible by 3.
Look at the following Series
3, 6, 9, 12, 15, 18, 123, 12342, ..?.
Digital root or digit sum of the series 3 is 3
6 is 6
9 is 9
12 is 3
15 is 6
81 is 9
123 is 6
12342 is 3
We are looking that the above numbers have digit sum 3, 6, 9, 3, 6 ,9, 6, 3 respectably.
So all the numbers with digit sum 3, 6, 9 are divisible by 3.
(ii) By Viloknam Method
if the sum of all the digits of the number is divisible by 3, then the number is divisible by 3.
6 is divisible by 3 then it is divisible by 3.
9 is divisible by 3 then it is divisible by 3
The sum of the number 12 is 1+2= 3, which is divisible by 3 then it is divisible by 3
The sum of the number 15 is 1+5= 6, which is divisible by 3 then it is divisible by 3
The sum of the number 81 is 8+1= 9, which is divisible by 3 then it is divisible by 3
The sum of the number 123 is 1+2+3= 6, which is divisible by 3 then it is divisible by 3
The sum of the number 12342 is 1+2+3+4+2= 12, which is divisible by 3 then 12342 is divisible by 3
Exercise 01
Find which of the following are divisible for not by 3.
01) 27
02) 54
03) 35
04) 60
05) 84
06) 77
07) 10
08) 75
09) 211
10) 124
11) 135
12) 310
13) 142
14) 413
15) 265
16) 600
17) 587
18) 859
19) 1285
20) 21570
21) 751283
22) 123456
23) 1234567
24) 12345675
25) 1234567890
Sol.
01) The digital sum of the number 27 is 2+7= 9, so that 27 is divisible by 3
or
01) The sum of the number 27 is 2+7= 9, which is divisible by 3 then 27 is divisible by 3
09) The digital sum of the number 211 is 2+1+1= 5, so that 211 is not divisible by 3
or
09) The sum of the number 211 is 2+1+1= 5, which is not divisible by 3 then 211 is not divisible by 3
19) The digital sum of the number 1285 is 1+2+8+5= 16 = 1+6 = 7, so that 1285 is not divisible by 3
or
19) The sum of the number 1285 is 1+2+8+5= 16, which is not divisible by 3.
The number 1285 is not divisible by 3
DIVISIBILITY BY 9
(i) By Digital Sum Method
All numbers with a digit sum of 9 are always divisible by 9.
Look at the following Series
9, 18, 1233, 123642
Digital root or digit sum of the series 9 is 9
18 = 1+8 = 9
1233=1+2+3+3=9
123642=1+2+3+6+4+2=1+8=9
We are looking that the above numbers have digit sum 9 respectably so all the numbers are divisible by 9.
(ii) By Viloknam Method
if the sum of all the digits of the number is divisible by 9, then the number is divisible by 9.
9 is 9
18 = 1+8 = 9
1233=1+2+3+3=9
123642=1+2+3+6+4+2=18
The sum of the number 9 is 9, which is divisible by 9.
The sum of the number 18 is 1+8= 9, which is divisible by 9 then it is divisible by 9.
The sum of the number 1233 is 1+2+3+3= 9, which is divisible by 9 then it is divisible by 9.
The sum of the number 123642 is 1+2+3+6+4+2= 18, which is divisible by 9 then 123642 is divisible by 9.
Exercise 02
Find which of the following are divisible for not by 9.
01) 27
02) 54
03) 35
04) 60
05) 84
06) 77
07) 10
08) 75
09) 211
10) 124
11) 135
12) 310
13) 142
14) 413
15) 265
16) 600
17) 587
18) 859
19) 1205
20) 21570
21) 751283
22) 123456
23) 1234567
24) 12345675
25) 1234567890
Sol.
01) Unit digit of 27 is 7 so that it is not divisible by 5.
03) Unit digit of 35 is 5 so that it is divisible by 5.
07) Unit digit of 211 is 1 so that it is not divisible by 5.
08) Unit digit of 19 is 9 so that it is not divisible by 2.
19) Unit digit of 1205 is 5 so that it is divisible by 5.
25) Unit digit of 1234567890 is 0 so that it is divisible by 5.
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