Prime Factorization Math Example 3

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Example 3

easy
Write the prime factorization of 8484 and use it to find all of its factors.

Solution

  1. 1
    Factor: 84=2×42=2×2×21=22×3×784 = 2 \times 42 = 2 \times 2 \times 21 = 2^2 \times 3 \times 7.
  2. 2
    Number of factors: (2+1)(1+1)(1+1)=12(2+1)(1+1)(1+1) = 12 factors.
  3. 3
    List all factors: 1,2,3,4,6,7,12,14,21,28,42,841, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.

Answer

84=22×3×784 = 2^2 \times 3 \times 7; factors: 1,2,3,4,6,7,12,14,21,28,42,841, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.
From the prime factorization p1a1⋯pkakp_1^{a_1} \cdots p_k^{a_k}, the total number of factors is (a1+1)(a2+1)⋯(ak+1)(a_1+1)(a_2+1)\cdots(a_k+1). Every factor is formed by choosing an exponent for each prime independently, within its allowed range.

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 4 medium

A rectangular garden can be arranged as a whole-number rectangle in exactly [formula] different ways

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