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".
Can you find some two-digit numbers that belong in two of the sets?
Can you find some two-digit numbers that belong in three sets?
What is the smallest number that belongs in all four sets?
How could you describe the pattern of the numbers that satisfy both Alison's and Sam's statements?
How about the numbers that satisfy both Alison's and Matt's statements?
Can you describe patterns for other pairs of statements?
2 digit numbers that belong in 2 of the sets:
All multiples of 10 are valid for alison and sams statement
so , 10,30,50etc
2 digit numbers that belong in 3 sets:
If you take alisons, beckie’s and matt’s principles.
You can see that 15 fits into all of them
If you take alison’s , sam’s and matt’s principles.
You would find that 30 fits into all of them
What's the smallest one that would fit into all of them
In order to solve this, we have to find a pattern or common number for each row
Alisons number:
Has to end with a five or a zero
Becky: triangle numbers
The formula is T=n(n+1)/2
1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,136,153,171,190,210
Sams principle: even , but not multiples of four
Matt:multiples of 3 but not nine:
Add up digits and cannot be divisible by nine
Sam: even, but not multiples of four
So the number that we have must end with a zero to fit alison's and sam's principle
The number has to be a triangle number that does not divide to nine
So we would need to find a “n†to plug into the formula that would divide to ten, so it would have to multiply to twenty
4 and 5 would give us 10 , but that's not divisible by 2
The next digit that ends with 10 is
15,16 which would give us 120, it's divisible by four
The next digit that ends with 10 is
20,21, this would give us 210 , that fits all the principles
So our number is 210
How could you describe the pattern of numbers that would satisfy both sam and allison's statements:
Since the last digit has to be even and a digit that is a multiple of 5, the last digit will have to be 0, but since the last two digits cannot be divisible by 4, in the last 2 digits, the first digit has to be an odd number and the last digit has to be 0
How about numbers that satisfy both alison and matt’s statements?
Alison needs it to be a multiple of 5, and matt needs all of the digits to add up and not be divisible by 9.
So the digits will have to end with a 5 or a zero, and the numbers have to sum up to something divisible by 3 but not 9
Eg:
45=4+5=9 33=3+3= 6
Cannot can