1) Can you find some two digit numbers that belong in two of the sets?
Alison and Becky: 10, 15, 45 (Multiple of 5, triangular number)
{In the Triangular numbers, these are the first three to appear as Multiples of 5}
Alison and Sam: 10, 30, 50 (Multiple of 5, even but not multiple of 4)
{Multiple of 5 and being even would be a multiple of 10, and not being a multiple of 4 would mean not being a multiple of 20}
Alison and Matt: 15, 30, 60 (Multiple of 5, multiple of three but not multiple of nine)
{Multiple of 5 and multiple of three would be multiples of 15, and not being a multiple of nine would mean not being a multiple of 45}
Becky and Sam: 6, 10, 66 (Triangular number, even but not multiple of four)
{In the Triangular numbers, these are the first three even numbers that aren’t a multiple of four}
Becky and Matt: 3, 6, 15 (Triangular numbers, multiple of three but not multiples of nine)
{In the Triangular numbers, these are the first three numbers that are a multiple of three but not nine}
Sam and Matt: 6, 30, 42 (Even but not multiple of four, multiple of three but not multiple of nine)
{Even and multiple of three would be multiples of 6, and not being multiples of four and nine would mean not being a multiple of 12 or 18 respectively}
2) Can you find some two digit numbers that belong in three sets?
Alison, Becky and Sam: 10, 190, 210
{In the Triangular numbers, these are the first three numbers to be both a multiple of five, and to be even, but not a multiple of four}
Alison, Becky and Matt: 15, 105, 120
{In the Triangular numbers, these are the first three numbers to be both a multiple of five, and a multiple of three but not 9}
Alison, Sam and Matt: 30, 150, 210
{Multiples of five, multiples of three and even numbers when grouped together would be multiples of 30, and not being a multiple of 4 or 9 would mean not being a multiple of 60 or 90}
Becky, Sam and Matt: 6, 66, 78
{In the Triangular numbers, these are the first three numbers to be both a multiple of three and two, while not being a multiple of nine or four}
3) What is the smallest number that belongs in all four sets?
210.
210/5=42 (Belongs in Alison)
210 is a triangular number (Belongs in Becky)
210 ends in 0, but 210/4=52.5 (Belongs in Sam)
210/3=70, but 210/9 has decimals (Belongs in Matt)
4) How could you describe the pattern of numbers that satisfy both Alison’s and Sam’s statements?
These numbers would have to be both a multiple of five and even. 5X2=10, so all multiples of 10 would count. Factor in that it can’t be a multiple of four, and multiples of 20 would be excluded. Therefore, the pattern would go 10, 30, 50, 70, 90 etc etc etc, adding 20 in between.
5) How about the numbers that satisfy both Alison’s and Matt’s statements?
These numbers would be both multiples of five and multiples of three but not nine. 5X3=15, so all multiples of 15 would count. Factor in that it can’t be a multiple of nine, and multiples of 45 would be excluded. Therefore, the pattern would go 15, 30, 60, 75, 105 etc etc etc, adding 15 then 30 and repeating that over and over.
6) Can you describe patterns for other pairs of statements?
Hopefully.
Alison and Becky:
These numbers would have to be both a triangular number, and a multiple of five. By looking at the sequence of triangular numbers, you would find that the first four of these numbers are the 4th, 5th, 9th and 10th triangular numbers. Continuing the pattern for the next four, this would be the 14th, 15th, 9th and 10th triangular numbers. Therefore, the nth triangular number, where n ends in 4, 5, 9 or 0 would count. The pattern would go 10, 15, 45, 55, 105 etc etc etc, with these numbers being the 4th, 5th, 9th, 10th and 14th term respectively.
Becky and Sam:
These numbers would have to be both a triangular number and even, but not a multiple of four. By looking at the sequence of triangular numbers, you would find that the first four of these would be the 3rd, 4th, 11th and 12th triangular number. Continuing the sequence, you would get the 19th, 20th, 27th and 28th triangular number. Therefore, by starting at three, and adding 1 whenever your current number is odd and adding 7 when it’s even, you would get all the terms that satisfy Becky and Sam’s statements. The pattern would go 6, 10, 66, 78, 190 etc etc etc, with these numbers being the 3rd, 4th, 11th, 12th and 19th term respectively.
Becky and Matt:
These numbers would have to be both a triangular number and a multiple of three, but not a multiple of nine. By looking at the sequence of triangular numbers, you would find that the first four of these would be the 2nd, 3rd, 5th and 6th triangular numbers. Continuing the sequence, you would get the 11th, 12th, 14th and 15th triangular numbers. Therefore, by starting at two, and adding one, then two, then one again and again, you would be almost done. Notice that if so, the 8th and 9th term are not there, even though 6+2=8 and 8+1=9. So by ignoring all terms that are a multiple of nine or that are one less than one, you would get all the terms that satisfy Becky and Matt’s statements. The pattern would go 3,6, 21, 28, 66 etc etc etc, with these numbers being the 2nd, 3rd, 5th, 6th and 11th term respectively.
Sam and Matt:
These numbers would have to be both a multiple of 2 and 3, but not multiples of 4 or 9. Therefore, these numbers would be a multiple of 6, but not a multiple of 12 or 18. You would find that the first couple of these numbers are 6, 42, 66, 78, 102, 114, 138. Therefore, the pattern would be to first add 36, and then to repeat alternatively adding 24 and adding 12. The first ten terms would be 6, 42, 66, 78, 102, 114, 138, 150, 174, 186.