Alison, Becky, Sam and Matt are playing a game.
Each of them writes down a statement that describes a set of numbers.
Alison writes "Multiples of five".
Becky writes "Triangular numbers".
Sam writes "Even, but not multiples of four".
Matt writes "Multiples of three but not multiples of nine".
1. Can you find some two-digit numbers that belong in two of the sets?
2. Can you find some two-digit numbers that belong in three sets?
3. What is the smallest number that belongs in all four sets?
4. How could you describe the pattern of the numbers that satisfy both Alison's and Sam's statements?
5. How about the numbers that satisfy both Alison's and Matt's statements?
6. Can you describe patterns for other pairs of statements?
1. I looked at the easier ones (The ones that requires the least working outs and thinking) for question 1. For example, Alison and Matt’s statements. The number required to fit both Alison and Matt’s statement is required to be a multiple of 5 and 3. Just times 5 and 3 together will get you 15. And according to Matt’s statement, 15 is not a multipe of 9.
2. For the second question, you can use the exact answer as the previous answer along with one more statement. So the previous answer is 15, which is a multiple of 5 and 3 but not a multiple of 9. We could use 15 is the answer to this question as well, but since I chose Sam’s statement as one of the three statements, that is not possible since it said that the answer must be even.
3. The number that is required to fit Alison and Sam’s statement is required to be a multiple of 5 and be even but no a multiple of 4. The first one is obviously 10, because 10 is a multiple of 5 and is even and is not a multiple of 4. The second one cannot be 15 because it is an odd number, not can it be 20 since it is a multiple of 4. You keep on finding multiples of 5 until you reach 30. It is a multiple of 5 and is not a multiple of 4 and is even. Afterwards, you keep on adding 5 to the previous number until you reach 50, which fits in with Alison and Matt’s statement. So those 70 and 90. So the numbers that fit to the statements are 10, 30, 50, 70, 90… The pattern is that each number goes up by 20 from the previous one.
4. The number that is required to fit Alison and Matt’s statements has to be a multiple of 5, and a multiple of 3 but not 9. The first one (According to my first answer) is 15, since it fit with the statements (Find 15 by multiplying 5 and 3 together). Since 15 is the smallest number that is a multiple of both 5 and 3, Th next number should be 15+15, which is 30. Which again fits all the required things to fit in with Alison and Matt’s statements. After that you add 15 to 30 again, which will be 45, which is NOT correct since it is a multiple of 9. Afterwards, add 15 again to get 60 which is correct. After, it will be 75, which is correct. But afterwards, we get 90 when we add 15 to 75, which is a multiple of 9 which is NOT correct. If you keep on adding 15 to the previous number. The formula for the pattern is x+15, except for the ones that has a multiple of 9 (45, 90, 135 etc)
5. For Alison and Becky’s statements, you are required a number that has a multiple of 5 and is a triangular number. (1,3,6,10,15…)First, we calcuate the triangular numbers from 1-100, which will be 1, 3, 6, 10, 15,21, 28, 36, 45, 55, 66, 78, 91. Afterwards, we can deduce the triangular numbers with multiples of 5. Which are 10, 15, 45, 55. You will then notice that the numebers that fit to the statements are on a certain position on the list of triangular numbers from 1-100. It comes after the 3rd triangular number and the 4th one. Here’s a more clear diagram: 1,3,6,10,15,21,28,36,45,55,66,78,91.