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".
Find two-digit numbers that belong in two sets : Alison and Sam - 10,30,50,70,90
How did I find this?
I found this because Alison’s numbers were all multiples of 5 and Sam had all even numbers which were not multiples of four, and because multiples of five in between each multiple of 10 are not even, the numbers had to be multiples of 10 in order to be even and still be a multiple of 5. I then listed all the multiples of ten up to 90, which were : 10,20,30,40,50,60,70,80,90 and then found out which numbers were divisible by four and eliminated them. The remaining numbers : 10,30,50,70 and 90 were the ones which fit into this criteria.
Find two-digit numbers that belong in three sets : Becky, Matt and Sam : 6,66,78
How did I find this?
I found this by listing out all the triangular numbers up to 100, which were 0,1,3,6,10,15,21,28,36,45,55,66,78,91. Then I found all the even numbers which were not multiples of 4 : 6,10,66,78. I then found out all the numbers from here which were multiples of three but not multiples of 9 : 6,66,78. Therefore the two-digit numbers which fit into the criteria of these three people are : 6,66 and 78.
The smallest number which belongs into all four sets : 210
How did I find this?
First I found a triangular number which was a multiple of 10 because of our rule : We had to find multiples of 10 in order to find a number which belonged in multiples of 5 and also even numbers which were not multiples of 4. These number could have been 10,190 and 210. 120 is not included because it is a multiple of 4. I then found that 190 and 10 were not multiple of 3, and therefore the smallest number which fits all the criteria is 210.
How could you describe the pattern of numbers that satisfy both Alison’s and Sam’s statements?
We could describe the pattern of numbers as starting from 10 then adding 20 each time. Or we could say 20n-10 because if we used this formula, for example the first number would be 10. We can prove this through the (first termx20)-10 which would be (1x20)-10 which would equal 10. Also we can prove this on the second term. (2x20)-10 would equal 30 which would also fit.
How about the numbers that satisfy both Alison’s and Matt’s statements?
You would have to multiply the multiple of 3 by 5 each time to find the next number in the sequence, with every third multiple being skipped. For example to find the first three numbers,15, 30 and 60, you would multiply the first multiple of 3, which is 3 by 5 which would equal 15. Then you would multiply the second of multiple of 3, which is 6 by 5 again which would equal 30. You would then skip 9 because it is the third multiple and go on to 12. 12 multiplied by 5 would equal 60.
Can you describe patterns for other pairs of statements?
Becky and Sam : I found the triangular numbers which matched up with even numbers which were not multiples of 4. The numbers were 6,10,66,78 which were the 3rd,4th,11th,12th,19th and 20th triangular numbers. From this I found that the term for the triangular numbers was 3rd term to start, the next term, the seventh term after the previous term and continuing like this.