Year 7 Sum of Primes/Squares/Cubes Puzzles
All problems © UKMT
1. [JMC 2015 Q11] What is the smallest prime number that is the sum of three different prime
numbers?
A 11
B 15
C 17
D 19
E 23
2. [JMO 1999 A2] In how many different ways can 50 be written as the sum of two prime
numbers? (Note: 𝑥 + 𝑦 and 𝑦 + 𝑥 do not count as different.)
3. [JMO 2009 A3] The positive whole numbers 𝑎, 𝑏 and 𝑐 are all different and
𝑎2 + 𝑏 2 + 𝑐 2 = 121. What is the value of 𝑎 + 𝑏 + 𝑐?
4. [JMC 2015 Q19] One of the following cubes is the smallest cube that can be written as the
sum of three positive cubes. Which is it?
A 27
B 64
C 15
D 216
E 512
5.
[JMC 2006 Q20] The sum of three different prime numbers is 40. What is the difference
between the two biggest of these numbers?
A 8
B 12
C 16
D 20
E 24
6. [JMC 2010 Q22] Kiran writes down six different prime numbers, 𝑝, 𝑞, 𝑟, 𝑠, 𝑡, 𝑢, all less than
20, such that 𝑝 + 𝑞 = 𝑟 + 𝑠 = 𝑡 + 𝑢. What is the value of 𝑝 + 𝑞?
A 16
B 18
C 20
D 22
E 24
7. [TMC Regional 2009 Q9] 12345 can be expressed as the sum of two primes in exactly one
way. What is the larger of the two primes?
 [JMO 2006 A9] The prime number 11 may be written as the sum of three prime numbers in
two different ways: 2 + 2 + 7 and 3 + 3 + 5. What is the smallest prime number which can
be written two different ways as the sum of the three prime numbers which are all
different?
 [JMO 2014 B6] The sum of four different prime numbers is a prime number. The sum of
some pair of the numbers is a prime number, as is the sum of some triple of the numbers.
What is the smallest possible sum of the four prime numbers?
Year 7 Sum of Primes/Squares/Cubes Puzzles
All problems © UKMT
1. [JMC 2015 Q11] What is the smallest prime number that is the sum of three different prime
numbers?
A 11
B 15
C 17
D 19
E 23
2. [JMO 1999 A2] In how many different ways can 50 be written as the sum of two prime
numbers? (Note: 𝑥 + 𝑦 and 𝑦 + 𝑥 do not count as different.)
3. [JMO 2009 A3] The positive whole numbers 𝑎, 𝑏 and 𝑐 are all different and
𝑎2 + 𝑏 2 + 𝑐 2 = 121. What is the value of 𝑎 + 𝑏 + 𝑐?
4. [JMC 2015 Q19] One of the following cubes is the smallest cube that can be written as the
sum of three positive cubes. Which is it?
A 27
B 64
C 15
D 216
E 512
5.
[JMC 2006 Q20] The sum of three different prime numbers is 40. What is the difference
between the two biggest of these numbers?
A 8
B 12
C 16
D 20
E 24
6. [JMC 2010 Q22] Kiran writes down six different prime numbers, 𝑝, 𝑞, 𝑟, 𝑠, 𝑡, 𝑢, all less than
20, such that 𝑝 + 𝑞 = 𝑟 + 𝑠 = 𝑡 + 𝑢. What is the value of 𝑝 + 𝑞?
A 16
B 18
C 20
D 22
E 24
7. [TMC Regional 2009 Q9] 12345 can be expressed as the sum of two primes in exactly one
way. What is the larger of the two primes?
 [JMO 2006 A9] The prime number 11 may be written as the sum of three prime numbers in
two different ways: 2 + 2 + 7 and 3 + 3 + 5. What is the smallest prime number which can
be written two different ways as the sum of the three prime numbers which are all
different?
 [JMO 2014 B6] The sum of four different prime numbers is a prime number. The sum of
some pair of the numbers is a prime number, as is the sum of some triple of the numbers.
What is the smallest possible sum of the four prime numbers?
Solutions (also on slides)
1.
2.
3.
4.
5.
6.
7.
8.
9.
D
4 ways (47 + 3, 43 + 7, 37 + 13, 31 + 19)
17
D
E
E
12343
23
31