mathematics

Sum-product number

Sum-product 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 Sum-product number rather than just read about it. In short: A sum-product number in a given number base b {\displaystyle b} is a natural number that is equal to the product of the sum of its digits and the product of its digits. There are a finite number of sum-product numbers in any given base b {\displaystyle b} .

Key takeaways

Sum-product 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 Sum-product number to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sum-product number from memory before moving on to harder problems.

Reference excerpt

A sum-product number in a given number base b {\displaystyle b} is a natural number that is equal to the product of the sum of its digits and the product of its digits. There are a finite number of sum-product numbers in any given base b {\displaystyle b} . In base 10, there are exactly four sum-product numbers (sequence A038369 in the OEIS): 0, 1, 135, and 144.

Definition Let n {\displaystyle n} be a natural number. We define the sum-product function for base b > 1 {\displaystyle b>1} , F b : N → N {\displaystyle F_{b}:\mathbb {N} \rightarrow \mathbb {N} } , to be the following:

F b ( n ) = ( ∑ i = 1 k d i ) ( ∏ j = 1 k d j ) {\displaystyle F_{b}(n)=\left(\sum _{i=1}^{k}d_{i}\right)\!\!\left(\prod _{j=1}^{k}d_{j}\right)}

where k = ⌊ log b ⁡ n ⌋ + 1 {\displaystyle k=\lfloor \log _{b}{n}\rfloor +1} is the number of digits in the number in base b {\displaystyle b} , and

d i = n mod b i + 1 − n mod b i b i {\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b}}^{i}}{b^{i}}}}

is the value of each digit of the number. A natural number n {\displaystyle n} is a sum-product number if it is a fixed point for F b {\displaystyle F_{b}} , which occurs if F b ( n ) = n {\displaystyle F_{b}(n)=n} . The natural numbers 0 and 1 are trivial sum-product numbers for all b {\displaystyle b} , and all other sum-product numbers are nontrivial sum-product numbers. For example, the number 144 in base 10 is a sum-product number, because 1 + 4 + 4 = 9 {\displaystyle 1+4+4=9} , 1 × 4 × 4 = 16 {\displaystyle 1\times 4\times 4=16} , and 9 × 16 = 144 {\displaystyle 9\times 16=144} . A natural number n {\displaystyle n} is a sociable sum-product number if it is a periodic point for F b {\displaystyle F_{b}} , where F b p ( n ) = n {\displaystyle F_{b}^{p}(n)=n} for a positive integer p {\displaystyle p} , and forms a cycle of period p {\displaystyle p} . A sum-product number is a sociable sum-product number with p = 1 {\displaystyle p=1} , and an amicable sum-product number is a sociable sum-product number with p = 2. {\displaystyle p=2.}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Sum-product number

Start with the simplest possible case. Write down what Sum-product 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 Sum-product 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 Sum-product 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 Sum-product number

In research: Sum-product 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 Sum-product 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: Sum-product number is common in secondary-school and first-year university syllabi. It links to neighbouring topics Arithmetic dynamics, Base-dependent integer sequences, so understanding it makes those chapters shorter.
In everyday life: Look for Sum-product 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 Sum-product number in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sum-product 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 Sum-product number out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Sum-product number in simple terms?

Why does Sum-product 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 Sum-product 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 Sum-product number.