What is the smallest three-digit number that is divisible by 6, by 8 and by 9?
(Total for Question 5 is 2 marks)
6
Is it possible for a number to be divisible by 3 and by 4 but not divisible by 2? Give a reason for your answer.
(Total for Question 6 is 1 mark)
7
Priya thinks of a number n. She finds that n leaves remainder 1 when divided by 3, remainder 2 when divided by 4, and remainder 3 when divided by 5. Work out the smallest positive value of n.
(Total for Question 7 is 2 marks)
8
Which of the following is a prime number?
A) 49
B) 51
C) 53
D) 55
E) 57
(Total for Question 8 is 1 mark)
9
Is there a digit x (0 to 9) that can be placed in 2x9 to make the three-digit number divisible by both 3 and 4? Explain your answer.
(Total for Question 9 is 1 mark)
10
Find the number between 200 and 300 that is divisible by 7, whose digits add to 10, and which is an odd number.
(Total for Question 10 is 2 marks)
11
Which of these numbers is divisible by 8?
A) 1235
B) 1232
C) 1237
D) 1239
E) 1243
(Total for Question 11 is 1 mark)
12
A four-digit number has digits 2, 3, 1 and d. The number is divisible by 9. Find d.
(Total for Question 12 is 1 mark)
13
Find the smallest positive integer greater than 10 that leaves remainder 2 when divided by both 4 and 6.
(Total for Question 13 is 1 mark)
14
Find a three-digit descending-consecutive digit number (digits like 7,6,5) that is divisible by 3 and by 5.
(Total for Question 14 is 1 mark)
15
Which of these numbers is divisible by both 2 and 3?
A) 143
B) 154
C) 156
D) 158
E) 160
(Total for Question 15 is 1 mark)
16
Find the largest integer less than 100 that has exactly the three distinct prime factors 2, 3 and 5 (so the number is of the form 2a * 3b * 5c with a,b,c ≥ 1).
(Total for Question 16 is 2 marks)
17
Which of these four-digit numbers is divisible by 11?
A) 3080
B) 2745
C) 4516
D) 3928
E) 6601
(Total for Question 17 is 1 mark)
18
Find the smallest number greater than 70 that ends with digit 1 and is divisible by 7.
(Total for Question 18 is 1 mark)
19
What is the smallest three-digit number that is divisible by 2, 3, 5 and 7?
(Total for Question 19 is 1 mark)
20
Find all two-digit numbers ab (a and b are non-zero digits and a ≠ b) such that the number ab is divisible by 4 and the reversed number ba is divisible by 3.
(Total for Question 20 is 3 marks)
Mark scheme · EP.M110 Divisibility Rules and Number Property Puzzles
Question 1
B1 correct option B (249) cao
Answer: B) 249 is divisible by 3 (2 + 4 + 9 = 15, which is a multiple of 3). (Answer: B) 249.)
Question 2
B1 correct option B (124) cao
Answer: B) 124 (last two digits 24, and 24 is divisible by 4).
Question 3
B1 correct option A (234) cao
Answer: A) 234 (2 + 3 + 4 = 9, which is divisible by 9).
Question 4
B1 correct option A (352) cao
Answer: A) 352 (alternating sum 3 - 5 + 2 = 0, which is divisible by 11).
Question 5
M1 LCM of 6, 8 and 9 found or implied (LCM = 72) oe
A1 144 cao
Answer: 144
Question 6
B1 No, because divisible by 4 implies the number is even (divisible by 2), so it cannot be not divisible by 2 cao
Answer: No. If a number is divisible by 4 it is even, so it must be divisible by 2.
Question 7
M1 Use of congruences or systematic search to show n = 20m + 18 or equivalent CRT working
A1 58 cao
Answer: 58
Question 8
B1 correct option C (53) cao
Answer: C) 53 (53 has no divisors other than 1 and 53).
Question 9
B1 No; divisible by 4 requires last two digits x9 to be divisible by 4, but any number ending in 9 is odd so cannot be divisible by 4 cao
Answer: No. A number ending in 9 is odd so cannot be divisible by 4; therefore no digit x makes 2x9 divisible by 4 (hence not by 12).
Question 10
M1 Show a candidate multiple of 7 in the 200s with digit sum 10 (eg 217) or systematic check of 7-multiples
A1 217 cao
Answer: 217
Question 11
B1 correct option B (1232) cao
Answer: B) 1232 (last three digits 232, and 232 รท 8 = 29).
Question 12
B1 d = 3 cao
Answer: 3
Question 13
B1 14 cao
Answer: 14
Question 14
B1 any correct example, e.g. 765 or 210 oe
Answer: 765 (or 210)
Question 15
B1 correct option C (156) cao
Answer: C) 156 (156 is even and 1 + 5 + 6 = 12, divisible by 3, so divisible by 6).
Question 16
M1 Consider multiples of 2*3*5 = 30 and multiply by powers until < 100
A1 90 cao
Answer: 90
Question 17
B1 correct option A (3080) cao
Answer: A) 3080 (3 - 0 + 8 - 0 = 11, divisible by 11).
Question 18
B1 91 cao
Answer: 91
Question 19
B1 210 cao
Answer: 210
Question 20
M1 Realise conditions: 10a + b = 0 (mod 4) and a + b = 0 (mod 3) (or equivalently 10b + a divisible by 3) shown
M1 Explain search or reason to reduce possibilities (consider two-digit multiples of 4 with non-zero digits)