mathematics

Reverse divisible number

Reverse divisible number is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Reverse divisible number rather than just read about it. In short: In number theory, reversing the digits of a number n sometimes produces another number m that is divisible by n. This happens trivially when n is a palindromic number; the nontrivial decimal reverse divisors are 1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, ...

Key takeaways

Reverse divisible number belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Reverse divisible number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Reverse divisible number from memory before moving on to harder problems.

Reference excerpt

In number theory, reversing the digits of a number n sometimes produces another number m that is divisible by n. This happens trivially when n is a palindromic number; the nontrivial decimal reverse divisors are

1089, 2178, 10989, 21978, 109989, 219978, 1099989, 2199978, ... (sequence A008919 in the OEIS). For instance, 1089 × 9 = 9801, the reversal of 1089, and 2178 × 4 = 8712, the reversal of 2178. The multiples produced by reversing these numbers, such as 9801 or 8712, are sometimes called palintiples.

Properties Every nontrivial decimal reverse divisor must be either 1/4 or 1/9 of its reversal. The number of d-digit nontrivial reverse divisors is 2 F ( ⌊ ( d − 2 ) / 2 ⌋ ) {\displaystyle 2F(\lfloor (d-2)/2\rfloor )} where F ( i ) {\displaystyle F(i)} denotes the ith Fibonacci number. For instance, there are two four-digit reverse divisors, matching the formula 2 F ( ⌊ ( d − 2 ) / 2 ⌋ ) = 2 F ( 1 ) = 2 {\displaystyle 2F(\lfloor (d-2)/2\rfloor )=2F(1)=2} .

History The reverse divisor properties of the first two of these numbers, 1089 and 2178, were mentioned by W. W. Rouse Ball in his Mathematical Recreations. In A Mathematician's Apology, G. H. Hardy criticized Rouse Ball for including this problem, writing:

"These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them which appeals to a mathematician. The proofs are neither difficult nor interesting—merely tiresome. The theorems are not serious; and it is plain that one reason (though perhaps not the most important) is the extreme speciality of both the enunciations and proofs, which are not capable of any significant generalization."

References

Worked examples

Example 1 — a first encounter with Reverse divisible number

Start with the simplest possible case. Write down what Reverse divisible number claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Reverse divisible number before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Reverse divisible number ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Reverse divisible number

In research: Reverse divisible number appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Reverse divisible number in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Reverse divisible number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Base-dependent integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Reverse divisible number outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Reverse divisible number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Reverse divisible number means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Reverse divisible number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Reverse divisible number in simple terms?

Why does Reverse divisible number matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Reverse divisible number?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Reverse divisible number.