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Counting factors from prime exponents

Counting factors from prime exponents

IP 1
8 marks
  1. Express \(360\) as a product of its prime factors in index notation.[1]
  2. Explain why every positive factor of \(360\) can be written as \(2^a\times3^b\times5^c\), where \(a\), \(b\) and \(c\) are whole numbers with \(0\le a\le3\), \(0\le b\le2\) and \(0\le c\le1\). Hence find the number of positive factors of \(360\).[3]
  3. Find the smallest positive integer that has exactly \(12\) positive factors.[2]
  4. Explain why a positive integer has an odd number of positive factors exactly when it is a perfect square.[2]

Solution:

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Answer:(a) \(360=2^3\times3^2\times5\) (b) \(24\) (c) \(60\) (d) factors pair as \(d\) and \(\dfrac nd\); only \(d=\sqrt n\) can be unpaired, which happens exactly when \(n\) is a perfect square

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