# DIVISIBILITY BY 7 - RULE WITH PROOF, SOLVED EXAMPLES

Multiplication Tables
and
Division before Divisibility By 7
They are prerequisites here.

Here we study methods that can be used to determine
whether a number is evenly divisible by Seven.

These are shortcuts for testing a number's factors
without resorting to division calculations.

The rules given below transform a given number's divisibility
by a divisor to a smaller number's divisibility by the same divisor.

If the result is not obvious after applying it once,
the rule should be applied again to the smaller number.

We present the rules with examples, in a simple way,
to follow, understand and apply. Great Deals on School & Homeschool Curriculum Books

Here is a collection of proven tips,
tools and techniques to turn you into
a super-achiever - even if you've never
thought of yourself as a "gifted" student.

and remember large chunks of information
with the least amount of effort.

If you apply what you read from the above
collection, you can achieve best grades without
giving up your fun, such as TV, surfing the net,
playing video games or going out with friends!

Know more about the Speed Study System.

## Rule for Divisibility By 7

To find out, if a number is divisible by 7, take the last digit, double it, and subtract it from the rest of the number. If you get an answer divisible by 7 (including zero), then the original number is divisible by 7.
If you don't know the new number's divisibility, you can apply the rule again.

If you want Proof of Rule for Divisibility By 7 ,
you may see at the bottom of the page.

Example 1 of Rule for Divisibility By 7

Find whether 343 is divisible by 7 or not.

Solution :
Twice the last digit = 2 x 3 = 6; Rest of the number = 34
Subtracting, 34 - 6 = 28 is divisible by 7. ( 28 = 4 x 7)
∴ 343 is divisible by 7. Ans.

Example 2 of Rule for Divisibility By 7

Find whether 8965 is divisible by 7 or not.

Solution :
Twice the last digit = 2 x 5 = 10; Rest of the number = 896
Subtracting, 896 - 10 = 886
To check whether 886 is divisible by 7 :
Twice the last digit = 2 x 6 = 12; Rest of the number = 88
Subtracting, 88 - 12 = 66 is not divisible by 7. ( 66 = 9 x 7 + 3)
∴ 8965 is not divisible by 7. Ans.

Example 3 of Rule for Divisibility By 7

Find whether 49875 is divisible by 7 or not.

Solution :
To check whether 49875 is divisible by 7 :
Twice the last digit = 2 x 5 = 10; Rest of the number = 4987
Subtracting, 4987 - 10 = 4977

To check whether 4977 is divisible by 7 :
Twice the last digit = 2 x 7 = 14; Rest of the number = 497
Subtracting, 497 - 14 = 483

To check whether 483 is divisible by 7 :
Twice the last digit = 2 x 3 = 6; Rest of the number = 48
Subtracting, 48 - 6 = 42 is divisible by 7. ( 42 = 6 x 7 )

∴ 49875 is divisible by 7. Ans.

Example 4 of Rule for Divisibility By 7

Find whether 987651 is divisible by 7 or not.

Solution :
To check whether 987651 is divisible by 7 :
Twice the last digit = 2 x 1 = 2; Rest of the number = 98765
Subtracting, 98765 - 2 = 98763

To check whether 98763 is divisible by 7 :
Twice the last digit = 2 x 3 = 6; Rest of the number = 9876
Subtracting, 9876 - 6 = 9870

To check whether 987 is divisible by 7 :
Twice the last digit = 2 x 7 = 14; Rest of the number = 98
Subtracting, 98 - 14 = 84

To check whether 84 is divisible by 7 :
Twice the last digit = 2 x 4 = 8; Rest of the number = 8
Subtracting, 8 - 8 = 0 is divisible by 7.

∴ 987651 is divisible by 7. Ans.

Example 5 of Rule for Divisibility By 7

Find whether 986953 is divisible by 7 or not.

Solution :
To check whether 986953 is divisible by 7 :
Twice the last digit = 2 x 3 = 6; Rest of the number = 98695
Subtracting, 98695 - 6= 98689

To check Divisibility By 7 of 98689 :
Twice the last digit = 2 x 9 = 18; Rest of the number = 9868
Subtracting, 9868 - 18 = 9850

To check Divisibility By 7 of 985 :Twice the last digit = 2 x5 = 10; Rest of the number = 98
Subtracting, 98 - 10 = 88

To check Divisibility By 7 of 88 :Twice the last digit = 2 x 8 = 16; Rest of the number = 8
Subtracting, 8 - 16 = -8 is not divisible by 7.

∴ 986953 is not divisible by 7. Ans.

## Research-based personalized Math Help tutoring program : Divisibility By 7

Here is a resource for Solid Foundation in
Math Fundamentals from Middle thru High School.
You can check your self by the

### FREE TRIAL.

Are you spending lot of money for math tutors to your
child and still not satisfied with his/her grades ?

Do you feel that more time from the tutor and
more personalized Math Help to identify and fix
the problems faced by your child will help ?

Here is a fool proof solution I strongly recommend
and that too With a minuscule fraction of the amount
you spent on tutors with unconditional 100% money
back Guarantee, if you are not satisfied.

### SUBSCRIBE, TEST, IF NOT SATISFIED, RETURN FOR FULL REFUND

It is like having an unlimited time from an excellent Tutor.

It is an Internet-based math tutoring software program
that identifies exactly where your child needs help and
then creates a personal instruction plan tailored to your
child’s specific needs.

If your child can use a computer and access
the Internet, he or she can use the program.
And your child can access the program anytime
from any computer with Internet access.

### Unique program to help improve math skills quickly and painlessly.

There is an exclusive, Parent Information Page provides YOU
with detailed reports of your child’s progress so you can
monitor your child’s success and give them encouragement.
These Reports include

• Time spent using the program
• Assessment results
• Personalized remediation curriculum designed for your child
• Details the areas of weakness where your child needs additional help
• Provides the REASONS WHY your child missed a concept
• List of modules accessed and amount of time spent in each module
• Quiz results
• Creates reports that can be printed and used to discuss issues with your child’s teachers
These reports are created and stored in a secure section
of the program, available exclusively to you, the parent.
The section is accessed by a password that YOU create and use.
No unauthorized users can access this information.

### Personalized remediation curriculum designed for your child

Thus The features of this excellent Tutoring program are

• Using detailed testing techniques
• Identifing exactly where a student needs help.
• Its unique, smart system pinpointing precise problem areas -
• slowly and methodically guiding the student
• raising to the necessary levels to fix the problem.

### Not a “one-size-fits-all” approach!

Its research-based results have proven that
it really works for all students! in improving
math skills and a TWO LETTER GRADE INCREASE in
math test scores!,if they invest time in using
the program.

Proven for More than 10,000 U.S. public school
students who increased their math scores.

## Proof of Rule for Divisibility By 7

Let 'D' ( > 10 ) be the dividend.

Let D1 be the units' digit
and D2 be the rest of the number of D.
i.e. D = D1 + 10D2

We have to prove
(i) if D2 - 2D1 is divisible by 7,
then D is also divisible by 7

and (ii) if D is divisible by 7,
then D2 - 2D1 is also divisible by 7.

Proof of (i) :
D2 - 2D1 is divisible by 7 ⇒ D2 - 2D1 = 7k where k is any natural number.
Multiplying both sides by 10, we get
10D2 - 20D1 = 70k
Adding D1 to both sides, we get
(10D2 + D1) - 20D1 = 70k + D1
⇒ (10D2 + D1) = 70k + D1 + 20D1
⇒ D = 70k + 21D1 = 7(10k + 3D1) = a multiple of 7.
⇒ D is divisible by 7. (proved.)

Proof of (ii) :
D is divisible by 7 ⇒ D1 + 10D2 is divisible by 7⇒ D1 + 10D2 = 7k where k is any natural number.
Subtracting 21D1 from both sides, we get
10D2 - 20D1 = 7k - 21D1
⇒ 10(D2 - 2D1) = 7(k - 3D1)
⇒ 10(D2 - 2D1) is divisible by 7
Since 10 is not divisible by 7,
(D2 - 2D1) is divisible by 7. (proved.)

In a similar fashion,
we can prove the divisibility rule for any prime divisor.

## Progressive Learning of Math : Divisibility By 7

Recently, I have found a series of math curricula
(Both Hard Copy and Digital Copy) developed by a Lady Teacher
who taught everyone from Pre-K students to doctoral students
and who is a Ph.D. in Mathematics Education.

This series is very different and advantageous
over many of the traditional books available.
These give students tools that other books do not.
Other books just give practice.
These teach students “tricks” and new ways to think.

These build a student’s new knowledge of concepts
from their existing knowledge.
These provide many pages of practice that gradually
increases in difficulty and provide constant review.

These also provide teachers and parents with lessons
on how to work with the child on the concepts.

The series is low to reasonably priced and include

Elementary Math curriculum

and

Algebra Curriculum.