## mathmath333 one year ago question

1. mathmath333

\large \color{black}{\begin{align} S=\{1,2,3,4,\cdots , 1000\} \hspace{.33em}\\~\\ \end{align}} How many arithmatic progressions can be formed from the elements of $$S$$ that start with $$1$$ and end with $$1000$$ and have at least $$3$$ elements.

2. anonymous

may u pls help me with my math?

3. anonymous

ok, let's work on this problem

4. anonymous

$$a_1=1$$, $$a_n=1000$$ let difference of terms be $$d$$

5. mathmath333

u mean common difference

6. anonymous

yeah

7. anonymous

Now we can calculate $$a_n$$

8. mathmath333

$$a_n=1+(n-1)d$$

9. anonymous

very good

10. anonymous

then, we have$999=(n-1)d$so $$d$$ must be a divisor of $$999$$

11. mathmath333

yeas

12. anonymous

ok, how many values $$d$$ can take?

13. anonymous

considering$999=3^3 \times 37$

14. anonymous

@mukushla can u help me pls

15. anonymous

@mathmath333 can u pls help me with my question

16. mathmath333

i didnt undestand about the value of d

17. anonymous

well$n-1=\frac{999}{d}$right?

18. mathmath333

yes

19. anonymous

since $$n-1$$ is an integer so $$\frac{999}{d}$$ must be an integer, right?

20. mathmath333

yes

21. anonymous

so d must be a divisor of 999

22. mathmath333

yes

23. anonymous

ok, how many positive divisors does 999 have?

24. mathmath333

i forgot how to do it ,do we add the powers of its prime divisors

25. anonymous

yeah, that right,$(3+1)(2+1)=8$which are$1, 3, 9, 27, 37, 111, 333, 999$

26. anonymous

how many of them are acceptable considering conditions of problem?

27. mathmath333

i m still in doubt how u got this (3+1)(2+1)

28. anonymous

sry, It's (3+1)(1+1)

29. mathmath333

ok np

30. anonymous

31. mathmath333

ans given is 7 , but how u got thaat

32. anonymous

well all of values for $$d$$ are acceptable, except $$d=1$$ which gives a value of $$n=2$$, and our arithmatic progressions have at least 3 elements

33. anonymous

for example when you let $$d=333$$ then $$n$$ becomes:$n-1=\frac{999}{333}=3$$n=4$

34. anonymous

which is acceptable

35. mathmath333

u checked all 8 options

36. anonymous

no need to check all of them because it's obvious that all of values give a $$n$$ that is acceptable

37. mathmath333

ohk thnx briliant/bright

38. anonymous

np, anytime math