Uses of Prime Factorisations
Note that you are permitted to use the ‘FACT’ button on your calculator.
Exercise 4 – Squares and Cubes
1. Given that 150 = 2 × 3 × 52 , what is the smallest number we need to multiply it by to make
it (a) a square number? (b) a cube number?
2. What is the smallest multiple of 16200 that is (a) square (b) cube.
3. What is the smallest number we need to multiply 24 by to make it (a) square (b) cube.
4. How many cube numbers are factors of 512 ?
5. What are the factors of 524 that are both square and cube? Leave your answer in factorised
index form.
6. [IMC 1999 Q16] On the right are three statements.
(i) 310 is even (ii) 310 is odd
Exactly which ones are true?
(iii) 310 is square
A (i) only
B (ii) only
C (iii) only
D (i) and (iii) only
E (ii) and (iii)
7. [JMO 2008 A6] How many positive square numbers are factors of 1600?
8. [SMC 2003 Q15] The number of this year, 2003, is prime. How many square numbers are
factors of 20032003 ?
9. [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 𝑥)
10. [Cayley 2013 Q1] What is the smallest non-zero multiple of 2, 4, 7 and 8 which is a square?
11. [Kangaroo Pink 2010 Q18] How many integers 𝑛, between 1 and 100 inclusive, have the
property that 𝑛𝑛 is a square number?
A 99
B 55
C 50
D 10
E 5
Exercise 5 – Number of Factors
1. [JMC 2012 Q3] Which of the following has exactly one factor other than 1 and itself?
A 6
B 8
C 13
D 19
E 25
2. By using the given prime factorisations, determine how many factors each of the following
numbers have.
a. 100 = 22 × 52
b. 45 = 32 × 5
c. 108 = 22 × 33
d. 210 = 2 × 3 × 5 × 7
3. How many factors do the following numbers have?
a. 107
b. 1510
4. A certain number has exactly eight factors including 1 and itself. Two of its factors are 33 and
15. What is the number?
5. How many numbers between 1 and 16 have an odd number of factors?
6. How many positive factors does 1212 have?
7. [JMO Mentoring Jun2011 Q2] How many positive divisors does 6! have including 6! and 1?
[6! = 6 × 5 × 4 × 3 × 2 × 1.]
8. [JMO 1997 B2] Every prime number has two factors. How many integers between 1 and 200
have exactly four factors?
9. [Junior Kangaroo 2015 Q23] How many three-digit numbers have an odd number of factors?
A 5
B 10
C 20
D 21
E 22
10. [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?
Exercise 6 – Trailing Zeroes
1. How many zeroes are at the end of 230 × 531 ?
2. How many zeroes are at the end of 240 × 330 × 520 ?
3. What is the last non-zero digit of:
a. 210 × 32 × 511
b. 28 × 57 × 72
c. 210 × 58 × 13
4. [JMO 2010 A3] Tom correctly works out 2010 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 34 × 45 × 56 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 257 × 34 × 553 is evaluated?
8. [Kangaroo Grey 2004 Q25] The number 𝑁 is the product of the first 100 positive whole
numbers. If all the digits of 𝑁 were written out, what digit would be next to all the zeros at
the end?
A 2
B 4
C 6
D 8
E 9
8
5
9. [SMC 2001 Q15] Sam correctly calculates the value of 5 × 8 . 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.
Answers
Section A
(a) 6 (b) 22 × 32 × 5 = 180
16200 = 23 × 34 × 52 (a) 24 × 34 × 52 = 32400 (b) 23 × 36 × 53 = 729000
24 = 23 × 3 (a) 2 × 3 = 6 (b) 32 = 9
5 (1, 53 , 56 , 59 , 512 )
1, 56 , 512 , 518 , 524
E
1600 = 26 × 52 . Factors are 1, 22 , 24 , 26 , 52 , 22 × 52 , 24 × 52 , 26 × 52. That’s 8.
1002
Numbers are 𝑥, 2𝑥, 6𝑥, 36𝑥 which add up to 45𝑥. Since 45 = 32 × 5, then 𝑥 could be 5 to
give 32 × 52 which is square. Thus Anastasia’s number was 5.
10. 784
11. B
1.
2.
3.
4.
5.
6.
7.
8.
9.
Section B
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
E (25)
(a) 9 (b) 6 (c) 12 (d) 16
(a) 64 (b) 121
165
Only square numbers have odd numbers of factors. There are four square numbers between
1 and 16.
1212 = 212 × 212 × 312 = 224 × 312 . Num factors = 25 × 13 = 325
30
Each number must be product of two distinct primes, or be a single prime number cubed, to
have four factors. There are 56 of the former, and 3 of the latter (8, 27, 125) giving 59
numbers.
E. If odd number of factors then number is square. Smallest three-digit square is 102 = 100.
Largest is 312 = 961. There are 31 − 10 + 1 = 22 squares between the two.
Each product must be 2 × 2𝑝 where 𝑝 is prime. Answer is 25
Section C
1. 31
2. 20
3. –
4.
5.
6.
7.
8.
9.
10.
11.
a. 5
b. 8
c. 2
10
20 = 210 × 1010 which finishes with 10 zeroes, so 11 digits.
2
6
6
4
11
1
1000 000 = 106 = 25 × 55 . We can’t group any 2s/5s in a single number. So we have 25 =
32 and 55 = 3125.