Maths WorkshopMaths problems with full solutions
Exercise 543Number theory › PrimesHigh schoolAdded 3 Nov 2026

The fundamental theorem of arithmetic

#fundamental theorem#factorization#uniqueness

Statement

State the fundamental theorem of arithmetic and factor 360360 into primes.

Hint

Divide out the smallest prime repeatedly.

Show solutionHide solution

Solution

Statement. Every integer n≥2n\ge2 can be written uniquely as a product of primes (up to order).

Factoring 360360.
360=23⋅32⋅5.360=2^3\cdot3^2\cdot5.

Check. 23=82^3=8, 32=93^2=9, 8⋅9⋅5=3608\cdot9\cdot5=360.

Conclusion. 360=23⋅32⋅5360=2^3\cdot3^2\cdot5.