Question

What is the divisibility rule for \[11,12\] and \[13\]?

Answer 1

Hint: From the question we have been asked the divisibility rules for \[11,12\] and \[13\]. So, for solving these kinds of questions we will explain the divisibility rules and then we will take a few examples and briefly we will discuss and simplify the divisibility rules so that we can understand the divisibility rules very clearly. So, we proceed with our solution as follows.

Complete step-by-step solution:
Divisibility Rule for \[11\]:
Divide the alternate digits in two different groups. Take the sum of alternate digits separately and find the difference of the two numbers. If the difference is \[0\] or is divisible \[11\], the number is divisible by \[11\].
Example: \[86456293\] is divided into two groups \[ \left\{ 8,4,6,9 \right\}\] and \[ \left\{ 6,5,2,3 \right\}\]. Sum of the groups is \[27\] and \[16\], whose difference is \[11\] and that is divisible by \[11\], \[86456293\] is divisible by \[11\].
Divisibility Rule for \[12\]:
If the number is divisible by both \[3\] and \[4\], the number is divisible by \[12\] .
Divisibility rule of \[3\] is that the sum of digits is divisible by \[3\] and the divisibility rule of \[4\] is that the last two digits are divisible by \[4\].
Example: In \[ 185176368\] the sum of all the digits is \[45\] and is divisible by \[3\] and also the last two digits \[68\] are divisible by \[4\]. As such the number \[ 185176368\] is divisible by \[12\].
Divisibility Rule for \[13\]:
Recall the divisibility rule of \[7\] , this works for \[13\] too. Starting from the right mark off the digits in groups of threes (just as we do when we put commas in large numbers).
Now add up an alternate group of numbers and find the difference between the two. If the difference is divisible by \[13\] , the entire number is divisible by \[13\].
For example: \[ 123448789113\] , these are grouped as \[ 123,448,789,113\] and \[123+789=912\] and \[448+113=561\].
As difference between \[912-561=351\]
As \[912-561=351\] is divisible by \[13\], the number \[123448789113\] is divisible by \[13\].

Note: Students should have good knowledge in the concept of divisibility rules. Students should know the divisibility rules of \[3\] and \[4\] to answer the question. We should not do any calculation mistakes like, in the case of example \[ 123448789113\] if we add the \[123+448\] instead of \[123+789=912\] the solution will be wrong as they are nt alternative according to the divisibility rule.