Question

Write the prime factorization of each number in exponential notation.

Answer 1

Answer:

1. (3^3)(5^2)(7)
2. (2^4)(3^2)(5)(7)(13)

Step-by-step explanation:

It is convenient to make use of divisibility rules as far as possible.

1. 4725 is obviously divisible by 25, so we have ...

  4725 = 5^2 × 189

The sum of digits of 189 is divisible by 9, so that number is as well

  4725 = 5^2 × 3^2 × 21

Your knowledge of multiplication tables tells you 21 = 3×7, so the prime factorization is ...

  4725 = 3^3 × 5^2 × 7

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2. 65520 is apparently divisible by 20, so we have ...

  65520 = 2^2 × 5 × 3276

3276 has a sum of digits divisible by 9, so it is divisible by 9.

  65520 = 2^2 × 3^2 × 5 × 364

364 is apparently divisible by 4, so ...

  65520 = 2^4 × 3^2 × 5 × 91

91 is divisible by 7, so the final prime factorization is ...

  65520 = 2^4 × 3^2 × 5 × 7 × 13

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Divisibility rules of use are

• divisible by 2 if ones digit is even
• divisible by 3 if sum of digits is divisible by 3
• divisible by 5 is ones digit is 0 or 5
• divisible by 7 if the number Nx is such that N-2x is divisible by 7 (N is the number with the ones digit (x) removed) Here, 91 ⇒ 9-2·1 = 7 is divisible by 7.
• divisible by 9 if sum of digits is divisible by 9