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Divisibility Checks: Tips & Rules

Upavidhi Blog on Mathematics | August 2016
 

A number, m, is said to be divisible by another number, n, if m ÷ n has no remainder. Although the most effective method to check for divisibility would seem to divide to find out, conventional mathematics provide a set of rules to find the results without executing the steps of division. These rules for divisibility checks come handy, especially in anticipating results.
 
Divisibility checks, in conventional mathematical methods, are based on rules and tricks that work only on specific situations. Let us refresh our memories, and start with the basic divisibility rules:
 
  • Divisibility by 2:
    If the last digit of a number is even (that is, the last digit is 0,2,4,6, or 8), then it is divisible by 2. For example, 15234 is divisible by 2 because the last digit is 4, which is even.
  • Divisibility by 3:
    If the sum of all the digits of a number is divisibile by 3, then it is divisible by 3. For example, 12345 is divisible by 3 because 1+2+3+4+5 = 15, which is divisible by 3.
  • Divisibility by 4:
    if the last two digits of a number forms a number that is divisible by 4, then it is divisible by 4. For example, 10316 is divisible by 4 because the last two digits, 16 is divisible by 4.
  • Divisibility by 5:
    If the last digit of a number is 5 or 0, then it is divisible by 5. For example, 10315 is divisible by 5 because the last digit is 5.
  • Divisibility by 6:
    If a number is divisible by both 2 and 3, then it is divisible by 6. For example, 12126 is divisible by 6 because it is divisible by 2 (the last digit is 6) and it is also divisible by 3 (1+2+1+2+6 = 12, divisible by 3).
  • Divisibility by 7:
    A number is divisible by 7, if double the last digit of the number when subtracted from the rest of the number is divisible by 7 (including 0). For example, 8645 is divisible by 7 because 5 × 2 = 10 and 864 - 10 = 854, is divisible by 7. In case of confusion, the process may be re-iterated.
  • Divisibility by 8:
    If the last 3 digits of a number forms a number that is divisible by 8, then the number is divisible by 8. For example, 10048 is divisible by 8 because 048 = 48, is divisible by 8.
  • Divisibility by 9:
    If the sum of all the digits of a number is divisibile by 9, then it is divisible by 9. For example, 32886 is divisible by 9 because 3+2+8+8+6 = 27, which is divisible by 9.
  • Divisibility by 10:
    If the last digit of a number is 0, then it is divisible by 10. For example, 98730 is divisible by 10 because the last digit is 0.
  • Divisibility by 11:
    A number is divisible by 11, if alternate adding and subtracting the digits from the left forms a number that is divisible by 11 (including 0). For example, 365167484 is divisible by 11 because 3-6+5-1+6-7+4-8+4 = 0.
  • Divisibility by 12:
    If a number is divisible by both 3 and 4, then it is divisible by 12. For example, 8376 is divisible by 12 because it is divisible by 3 (8+3+7+6 = 24, divisible by 3) and it is also divisible by 4 (76 is divisible by 4).
  • Divisibility by 13:
    A number is divisible by 13, if 9 times the last digit of the number when subtracted from the rest of the number is divisible by 13 (including 0). For example, 4797 is divisible by 13 because 9 × 7 = 63 and 479 - 63 = 416, is divisible by 7. In case of confusion, the process may be re-iterated.
There are more such rules, but alas, not for all numbers. Each rule work only for specific numbers, and they tend to get very cumbersome. For example, the rule for 17, makes one use the alternating sums of 8-digit groups!
 
As we dwelve into the rules, it becomes obvious that the rules above are only specific to numbers and it keeps getting complex for larger numbers. With the absence of a proper technique in conventional mathematics, we are left with no other option than to perform a division to check for divisibility.
 
This is one of the areas of mathematics where Vedic Mathematics comes to the rescue.
 
It is important to note, that division by 2 (and even 3), 5 and 10 are relatively very easy and maybe executed mentally. So, the problem of divisibility checks is reduced to odd-numbers, ending with 1, 3, 7, and 9. For example, if we are to know if 304 is divisible by 38, we can divide both the numbers by 2 to reach the underlying problem: if 152 is divisible by 19.
 
Vedic Mathematics implements a technique known as veṣṭanaṃ, or osculation (discussed here ») that reduces the problem into a series of small additions and multiplications to mentally check for divisibility. For example, to check divisibility of 152 by 19:
  The positive osculator for 19 is 2.
(Drop the last 9, and add 1 to the remaining number)
 
Add, and multiply with osculator till the digits are exhausted. If the result is divisible by 19, so is the original number, 152.
In this case,
(2 × 2) + 5 = 9
And, (9 × 2) + 1 = 19
 
Thus, 152 is divisible by 19
This technique is not only unique, it provides a generalized mechanism for divisibility checks - unfound in conventional mathematics, with a marked advantage that we do not have to remember the innumerous rules for divisibility checks for each number. Let us take another example, to check divisibility of 168 by 11:
  The negative osculator for 11 is 1.
(Drop the last 1, and keep remaining number)
 
Re-arrange the number as alternate negative and positive digits, 168
 
Add, and multiply with osculator till the digits are exhausted. If the result is divisible by 11, so is the original number, 168.
In this case,
(8 × 1) + 6 = 2
And, (2 × 1) + 1 = 3
 
Thus, 168 is not divisible by 11
The example above, seems very similar to the "Divisibility by 11" rule, but provides a far-more generalized approach. For example, to check divisibility of 168 by 21:
  The negative osculator for 21 is 2.
(Drop the last 1, and keep remaining number)
 
Re-arrange the number as alternate negative and positive digits, 168
 
Add, and multiply with osculator till the digits are exhausted. If the result is divisible by 11, so is the original number, 168.
In this case,
(8 × 2) + 6 = 10
And, (10 × 2) + 1 = 21
 
Thus, 168 is divisible by 21
As is typical to Vedic Mathematics, these steps are relatively simple and may be executed mentally, with some practice. This technique drastically reduces the burden of remembering rules for divisibility checks, and certainly provides a fool-proof mechanism to check for divisibility without performing a division. The detailed mathematical technique is discussed here »
 
To conclude, knowledge of Vedic Mathematics indeed works wonders - especially in speeding up the operations, eradicating chances of silly mistakes, and most importantly, in helping one to visualize the entire process in a simple, transparent manner. These advantages, in turn, allows oneself to concentrate on the overall mathematical problem.
 
However, in line with our stand at Upavidhi, it should only be viewed as an extension to the conventional methods, and applied only after expertise - that comes though practice, and a certain amount of mathematical skill.
 
« July 2016
 
 
 
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