Set- i do not understand the question

[icystrike]

Homework Statement

What is the maximum size of a subset, P, of {1, 2, 3, . . . , 50}
with the property that no pair of distinct elements of P
has a sum divisible by 7?

Homework Equations

The Attempt at a Solution

smallest sum = 3
largest sum = 99

factors of 7:

7 14 21 28 35 42 49 56 63 70 77 84 91 98

7 can be form by:

1 6
2 5
3 4

 

[tiny-tim]

Hi icystrike!

What is the maximum size of a subset, P, of {1, 2, 3, . . . , 50}
with the property that no pair of distinct elements of P
has a sum divisible by 7?
…
7 can be form by:

1 6
2 5
3 4

ok … so if 1 is in P, what can't be in P?

 

[icystrike]

1 2 3 4 5 7 8 9 10 11 12 14 15 16 17 18 19 21 22 23 24 25 26 28 29 30 31 32 33 35 36 37 38 39 40 42 43 44 45 46 47 49 50 !
yes

 

[tiny-tim]

1 2 3 4 5 7 8 9 10 11 12 14 15 16 17 18 19 21 22 23 24 25 26 28 29 30 31 32 33 35 36 37 38 39 40 42 43 44 45 46 47 49 50 !
yes

no, that's what could be in P … what can't be in P are 6 13 20 27 34 41 48, in other words everything = -1 (mod 7)

suppose 2 and 3 are also in P (as well as 1) … now what can't be in P?

 

[icystrike]

no, that's what could be in p … what can't be in p are 6 13 20 27 34 41 48, in other words everything = -1 (mod 7)

suppose 2 and 3 are also in p (as well as 1) … now what can't be in p?

5 12 19 26 33 40 47
4 11 18 25 32 39 46

 

[tiny-tim]

5 12 19 26 33 40 47
4 11 18 25 32 39 46

ok, so if 1 2 and 3 are in, then all the mod6s 5s and 4s are out …

can the other mod 1s 2s and 3s be in?