Uses of Prime Factorisations

Note that you are permitted to use the ‘FACT’ button on your calculator.

Exercise 4

  1. Use the following prime factorisations (if provided) to find the smallest number we need to multiply the number by to make it (a) a square (b) a cube.
  2. 16200
  3. How many cube numbers are factors of ?
  4. Use the fact that to list out all the factors of 12 (excluding 1) in prime factorised form.
  5. By using the given prime factorisations, determine how many factors each of the following numbers have.
  6. [JMC 2012 Q3] Which of the following has exactly one factor other than 1 and itself?
    A 6B 8C 13D 19E 25
  7. [IMC 1999 Q16] On the right are three statements.(i) is even (ii) is odd
    Exactly which ones are true?(iii) is square
    A (i) onlyB (ii) onlyC (iii) onlyD (i) and (iii) onlyE (ii) and (iii)
  8. [JMO 2008 A6] How many positive square numbers are factors of 1600?
  9. How many factors do the following numbers have?
  10. How many numbers between 1 and 16 have an odd number of factors?
  11. [Junior Kangaroo 2015 Q23] How many three-digit numbers have an odd number of factors?
    A 5B 10C 20D 21E 22
  12. [JMO 1997 B2] Every prime number has two factors. How many integers between 1 and 200 have exactly four factors?
  13. [Cayley 2013 Q1] What is the smallest non-zero multiple of 2, 4, 7 and 8 which is a square?
  14. [JMO 2012 B2] Anastasia thinks of a positive integer, which Barry then doubles. Next, Charlie trebles Barry's number. Finally, Damion multiplies Charlie's number by six. Eve notices that the sum of these four numbers is a perfect square. What is the smallest number that Anastasia could have thought of? (Hint: make Anastasia’s number )

Supplementary Questions

  1. What are the factors of that are both square and cube? Leave your answer in factorised index form.
  2. How many positive factors does have?
  3. A certain number has exactly eight factors including 1 and itself. Two of its factors are 33 and 15. What is the number?
  4. [SMC 2003 Q15] The number of this year, 2003, is prime. How many square numbers are factors of ?
  5. [Kangaroo Pink 2010 Q18] How many integers , between 1 and 100 inclusive, have the property that is a square number?
    A 99B 55C 50D 10E 5
  6. [JMO Mentoring Jun2011 Q2] How many positive divisors does 6! have including 6! and 1? [.]
  7. [JMO 1996 B1] How many positive whole numbers up to and including 400 can be written in exactly one way as the product of two even numbers?

BONUS EXERCISE (at end of slides) – Trailing Zeroes

  1. How many zeroes are at the end of ?
  2. How many zeroes are at the end of
  3. What is the last non-zero digit of:
  4. [JMO 2010 A3] Tom correctly works out and writes down his answer in full. How many digits does he write down in his full answer?
  5. [IMC 2007 Q7] If the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are all multiplied together, how many zeros are at the end of the answer?
  6. [IMC 2000 Q18] The number is written out in full. How many zeroes are there at the end of the number?
  7. [Kangaroo Pink 2012 Q16] What is the last non-zero digit when is evaluated?
  8. [Kangaroo Grey 2004 Q25] The number is the product of the first 100 positive whole numbers. If all the digits of were written out, what digit would be next to all the zeros at the end?
    A 2B 4C 6D 8E 9
  9. [SMC 2001 Q15] Sam correctly calculates the value of . How many digits does her answer contain?
  10. [Senior Kangaroo 2012 Q1] How many zeroes are there at the end of the number which is the product of the first 2012 prime numbers?
  11. Find two integers, neither of which has a zero digit, whose product is 1 000 000.