help

[z51]

Zenonas observed that the number x = 2018 has the following properties:
x is a multiple of 2.
x+1 is a multiple of 3. x+2 is a multiple of 4.
The number of positive integers, that are less than 2018 and satisfy the above properties, is:
Α. 100 Β. 112 Γ. 120 Δ. 168 Ε. 180

I didnt even know how to start so I tryed to run a program on c++ maybe to find something common in these numbers. But i dont know how to solve it mathematically. the answer is 168 btw

 

[Dr.Peterson]

Zenonas observed that the number x = 2018 has the following properties:
x is a multiple of 2.
x+1 is a multiple of 3. x+2 is a multiple of 4.
The number of positive integers, that are less than 2018 and satisfy the above properties, is:
Α. 100 Β. 112 Γ. 120 Δ. 168 Ε. 180

I didnt even know how to start so I tryed to run a program on c++ maybe to find something common in these numbers. But i dont know how to solve it mathematically. the answer is 168 btw

You haven't indicated how much you know about number theory (or what course this is for); the question can be easily stated in term of modular arithmetic.

But you can find the answer without any of that.

What is the smallest number that fits the description?

How much would you have to add to any such number to get to the next.

Then a division and a little double-checking, and you get the answer.

Give it a try!

 

[Steven G]

2 3 4 X
4 5 6
6 7 8
8 9 10
10 11 12
12 13 14
14 15 16 X
16 17 18
18 19 20
20 21 22
22 23 24
24 25 26
26 27 28 X
28 29 30
30 31 32
32 33 34
34 35 36
36 37 38
38 39 40 X


Do you see a pattern??