1) This number must be divisible by both 5 and 6.
A few examples are 30, 60 and 90.
The smallest number which is a multiple of both 5 and 6 is 30.
It helps to use the number sieve to show the multiples of 5 and then also allow it to show the multiples of 6. The remaining white circles are divisible by both 5 and 6. The smallest number which is white, is the smallest number which is a multiple of 5 and 6.
2) This number must be divisible by 4, 5 and 6.
A few examples are 60, 180 and 210.
The smallest number which is a multiple of 4, 5 and 6 is 60.
Again, use the same method as Question 1, however, also show the multiples of 4.
3)
a) Yes, a good example would be 57.
(7 x 8) + 1 = 57
(4 x 14) + 1 = 57
b) Yes, a good example would be 28.
(5 x 5) + 3 = 28
(10 x 2) + 8 = 28
c) No
For Question 3, use the 'Reminder' feature on the number sieve.
For example, 'a number that is 1 more than a multiple of 7' is a number 'divisible by 7 with remainder 1'.
Similar to Question 2, any white numbers after the two conditions have been added will be the numbers which are the same that the friends may be thinking.
Extension:
This question was tricky at first.
The group of pupils started by added to the number sieve:
'divisible by 2, remainder 1'
'divisible by 3, remainder 2'
...
'divisible by 10, remainder 9'
adding another condition in order as they made their way down the list.
They noticed a clear pattern forming which each addition, with every new 'remainder' added, there was a clear sequence in white numbers showing.
As a condition was added, the number of white circles went down and the gap between each white number went up but it wasn't random.
All white numbers fell on the same column. (Pattern forming is shown in photos)
The pupils couldn't find a number because the number sieve was unresponsive for large numbers but all pupils agreed that with infinite numbers, eventually, there would be a number that satisfies all the conditions.
On a faster computer, I was able to find some of those numbers (went up to 100000).
The smallest was 2519.
Another example is 5039.
Darcey, Oliver, Freddie, James and Anna - Year 6 Pupils