Prime Factorization Math Example 4

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Example 4

medium
A rectangular garden can be arranged as a whole-number rectangle in exactly 66 different ways (including 1×n1 \times n and n×1n \times 1 as different). What is the smallest possible area for this garden?

Solution

  1. 1
    The number of factor pairs (ordered, including 1×n1 \times n) equals the total number of factors. We need a number with exactly 66 factors.
  2. 2
    For 6 factors, the exponent pattern can be: (5)(5) giving p5p^5, or (2,1)(2,1) giving p2qp^2 q (factors: (2+1)(1+1)=6(2+1)(1+1)=6).
  3. 3
    Smallest p5=25=32p^5 = 2^5 = 32. Smallest p2q=22×3=12p^2 q = 2^2 \times 3 = 12.
  4. 4
    Smallest is 1212: factors 1,2,3,4,6,121, 2, 3, 4, 6, 12 — six factors. ✓

Answer

The smallest area is 1212 square units.
The number of factors of n=p1a1p2a2⋯n = p_1^{a_1}p_2^{a_2}\cdots is (a1+1)(a2+1)⋯(a_1+1)(a_2+1)\cdots. To minimise the value while achieving exactly 66 factors, we assign larger exponents to smaller primes — the classic 'smallest number with kk factors' optimisation.

About Prime Factorization

Writing a whole number as a product of prime numbers; every composite number has exactly one such representation (up to order).

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More Prime Factorization Examples

Example 1 easy

Find the prime factorization of [formula] using a factor tree.

Example 2 hard

Use prime factorization to find [formula] and [formula].

Example 3 easy

Write the prime factorization of [formula] and use it to find all of its factors.

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