Divisibility Problems
1.
[J31] Every digit of a given positive integer is either a 3 or a 4 with each occurring at least once. The
integer is divisible by both 3 and 4. What is the smallest such integer?
2. [J50] The eight-digit number “ppppqqqq”, where p and q are digits, is a multiple of 45. What are the
possible values of p?
3. [M07] (a) A positive integer N is written using only the digits 2 and 3, with each appearing at least
once. If N is divisible by 2 and by 3, what is the smallest possible integer N?
(b) A positive integer M is written using only the digits 8 and 9, with each appearing at least once. If M
is divisible by 8 and by 9, what is the smallest possible integer M?
4. [M55] A palindromic number is one which reads the same when its digits are reversed, for example
23832. What is the largest six-digit palindromic number which is exactly divisible by 15?
5. [J16] Find a rule which predicts exactly when five consecutive integers have sum divisible by 15.
6. [M96] Find the possible values of the digits p and q, given that the five-digit number ‘p543q’ is a
multiple of 36.
7. [M127] The five-digit number ‘𝑎679𝑏’, where 𝑎 and 𝑏 are digits, is divisible by 36. Find all possible
such five-digit numbers.
8. [M31] Find the smallest positive multiple of 35 whose digits are all the same as each other.
9. Show that:
a. 𝑛3 − 𝑛 is divisible by 6 for all integers 𝑛.
b. 𝑛4 + 2𝑛3 − 𝑛2 − 2𝑛 is divisible by 24 for all integers 𝑛.
10. [Based on NRich] If the digits 5, 6, 7 and 8 are inserted at random in 3_1_4_0_92 (one in each space),
what is the probability that the number created will be a multiple of 396 if:
a. Each of 5, 6, 7, 8 is used exactly once in each of the four gaps.
b. Each of 5, 6, 7, 8 can be used multiple times.
Divisibility Problems
1.
[J31] Every digit of a given positive integer is either a 3 or a 4 with each occurring at least once. The
integer is divisible by both 3 and 4. What is the smallest such integer?
2. [J50] The eight-digit number “ppppqqqq”, where p and q are digits, is a multiple of 45. What are the
possible values of p?
3. [M07] (a) A positive integer N is written using only the digits 2 and 3, with each appearing at least
once. If N is divisible by 2 and by 3, what is the smallest possible integer N?
(b) A positive integer M is written using only the digits 8 and 9, with each appearing at least once. If M
is divisible by 8 and by 9, what is the smallest possible integer M?
4. [M55] A palindromic number is one which reads the same when its digits are reversed, for example
23832. What is the largest six-digit palindromic number which is exactly divisible by 15?
5. [J16] Find a rule which predicts exactly when five consecutive integers have sum divisible by 15.
6. [M96] Find the possible values of the digits p and q, given that the five-digit number ‘p543q’ is a
multiple of 36.
7. [M127] The five-digit number ‘𝑎679𝑏’, where 𝑎 and 𝑏 are digits, is divisible by 36. Find all possible
such five-digit numbers.
8. [M31] Find the smallest positive multiple of 35 whose digits are all the same as each other.
9. Show that:
a. 𝑛3 − 𝑛 is divisible by 6 for all integers 𝑛.
b. 𝑛4 + 2𝑛3 − 𝑛2 − 2𝑛 is divisible by 24 for all integers 𝑛.
10. [Based on NRich] If the digits 5, 6, 7 and 8 are inserted at random in 3_1_4_0_92 (one in each space),
what is the probability that the number created will be a multiple of 396 if:
a. Each of 5, 6, 7, 8 is used exactly once in each of the four gaps.
b. Each of 5, 6, 7, 8 can be used multiple times.