No matter whether you liked or hated mathematics in your school, I can say you must have loved fancying about knowing the divisibility tests of 2, 3, 5 & 10. And if you were among the brainy students you also knew the divisibility tests of 4, 6, 8 & 11. But, what about the divisibility test of 7? Do you know about it? In today’s article, we’ll talk about the previous and newly discovered divisibility tests of 7.

The divisibility of a number by 7 is not as easy as the other divisibility test. In order to find it out, you need to follow a couple of recursive steps. Most importantly, there has been a divisibility test of 7 with well-defined steps. But the recent discovery by Chika Ofili about the same has been a hot topic in the headlines.

**The Foregoing Test**

**Step 1:** Separate the last digit of the number.

**Step 2:** Double the last digit and subtract from the remaining number.

**Step 3:** Repeat the steps unless you get a number within 0-70.

**Step 4:** If the result is divisible by 7, the number you started with is also divisible by 7.

**About Chika’s discovery**

A 12-year old Nigerian boy, Chika Ofili, made history this year after his new discovery in the field of mathematics. He was awarded at the TruLittle Hero Awards for discovering the new divisibility test of 7, popularly called as Chika’s Test.

Miss Mary Ellis, Ofili’s teacher and the head of the Mathematics Department of Westminster Under School, said that she gave all the students a book called First Steps for Problem Solvers (published by the UKMT) to study during the holidays. The book had several divisibility tests but it had no memorable test for 7. After the vacations, Ofili came to her saying that he had discovered a formula for the divisibility of 7 and also had the algebraic proof to back it up.

**Chika’s Test**

It is similar to the previously foregoing rule, but it is simpler and faster than it.

**Step 1:** Separate the last digit of the number.

**Step 2:** Multiply the last digit by 5 and add it to the remaining number.

**Step 3:** Repeat the steps unless you get a number within 0-70.

**Step 4:** If the result is divisible by 7, the number you started with is also divisible by 7.

**Proof of Chika’s Test**

Honestly, I am not sure about the proof Chika had to prove his discovery. This is just a supporting proof of the working rules and credibility of Chika’s Test.

**Let us say, a number N is divisible by 7.*** ( N = 7K, where K is an integer )*

N = 7K = 10^{n}a_{n}
+ 10^{n-1}a_{n-1} + 10^{n-2}a_{n-2} + 10^{n-3}a_{n-3}
+ ….. + 100a_{2} + 10a_{1} + a_{0}

=
10[ 10^{n-1}a_{n} + 10^{n-2}a_{n-1} + 10^{n-3}a_{n-2}
+ ….. + 10a_{2} + a_{1}] + a_{0}

**Adding and subtracting ‘50a**_{0}**’ on the RHS .**

N = 10[ 10^{n-1}a_{n}
+ 10^{n-2}a_{n-1} + 10^{n-3}a_{n-2} + ….. + 10a_{2}
+ a_{1}] + 50a_{0} -50a_{0} + a_{0}

N = 10[ 10^{n-1}a_{n}
+ 10^{n-2}a_{n-1} + 10^{n-3}a_{n-2} + ….. + 10a_{2}
+ a_{1} + 5a_{0 }] – 50a_{0} + a_{0}

N = 7K = 10[ 10^{n-1}a_{n} + 10^{n-2}a_{n-1} + 10^{n-3}a_{n-2} + ….. + 10a_{2} + a_{1} + 5a_{0 }] – 49a_{0}

7K + 49a_{0} = 10[ 10^{n-1}a_{n}
+ 10^{n-2}a_{n-1} + 10^{n-3}a_{n-2} + ….. + 10a_{2}
+ a_{1} + 5a_{0 }]

**Let P = 10**^{n-1}**a**_{n}** + 10**^{n-2}**a**_{n-1}** + 10**^{n-3}**a**_{n-2}** + ….. + 10a**_{2}** + a**_{1}** + 5a**_{0}

7K + 49a_{0} = 10P

Now, LHR : (7K + 49a_{0}) ≡ 0 (mod 7) => RHS : 10P ≡ 0 (mod 7)

Also, 10 ≡ 3 (mod 7) => for RHS : 10P ≡ 0 (mod 7) , P ≡ 0 (mod 7)

**Therefore, this proves Chika’s Test working rule and credibility.**

Well, you may say how is this beneficial or faster? Yes, it is better than the foregoing test, reason being,

- multiplying by 5 helps to reach a number within 0-70 at a faster rate compared to multiplication by 2.
- adding two numbers is psychologically simpler than subtraction.

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Did the child provide the proof of his algorithm as well? What I intend to know, is if he got the result by a trial-and-error approach, or had a well-informed hunch about his test.

Hello Chandrashekar!!

The Algorithm was completely based on Chika’s Observation and Trial-and-Error Approach. Prof. Simon Ellis (the younger brother of Mary Ellis), who also taught mathematics, tested and wrote an algebraic proof for the theory proposed by Chika.

The proof of this test seems to allow false positives… Ie numbers which aren’t divisible by 7 but still satisfy the test. Because it’s already assuming the number to be divisible by 7. Is there a complete proof?

Can we not consider it as positive osculator for 7, in opposition to the already present and well established concept of the negative osculator of 7 which is given as negative 2?

Hey Alex, nice observation, ideally the last digit is being multiplied by a positive osculator (+5) and added to the rest. But, I am not sure where Chika knew about the Osculator Method of checking divisibility.

[…] This has come to be called “Chika’s Test”. It might be faster than my other method, because it is faster to get to a recognizable multiple of 7 ( a number between 0 and 70) by adding multiples of 5 than subtracting multiples of 2. And adding numbers is sometimes simpler than subtracting numbers. Read more about Chika’s test here. […]