Every 7th Fibonacci number is divisible by 13.

Quick Scoop: The Core Answer

  • In the Fibonacci sequence, terms go: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …
  • If you look at every 7th term (7th, 14th, 21st, …), each of those terms is a multiple of 13.
  • The 7th Fibonacci number itself is 13, which is already a multiple of 13, and this pattern keeps repeating at regular intervals.

So, the direct answer to:

every 7th element in the Fibonacci sequence is divisible by what number?

is:

Every 7th Fibonacci number is divisible by 13.

Why 13 Shows Up

Mathematicians study Fibonacci numbers modulo (i.e., “remainders with”) different integers and find repeating patterns called periods. For modulus 13, the pattern lines up so that Fibonacci numbers that land at positions 7, 14, 21, … are all congruent to 0 modulo 13, meaning they are divisible by 13.

A small sample:

  • F7=13F_7=13F7​=13 → divisible by 13
  • F14=377=13×29F_{14}=377=13\times 29F14​=377=13×29 → divisible by 13
  • F21=10946=13×842F_{21}=10946=13\times 842F21​=10946=13×842 → divisible by 13

Related Cool Patterns

This property is part of a larger family of divisibility patterns in Fibonacci numbers:

  • Every 3rd Fibonacci number is divisible by 2.
  • Every 4th Fibonacci number is divisible by 3.
  • Every 5th Fibonacci number is divisible by 5.
  • Every 6th Fibonacci number is divisible by 8.
  • Every 7th Fibonacci number is divisible by 13.
  • Every 8th Fibonacci number is divisible by 21.

These patterns are a classic topic in recreational and school-level mathematics and often appear in problem sets and fun fact lists.

SEO-style extras

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  • Meta-style summary: Every 7th Fibonacci number is divisible by 13, a neat divisibility pattern that sits alongside other regular divisibility results in the Fibonacci sequence.

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