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# 8. Show that if (1 02 03, < 04 C5 06 are natural numbers; then 9 | (aj Ci _ Or 9 | (aj + 4i) , for some pair G; # aj- Hint: Pigeon hole principle: Approach 1: co...

## Question

###### 8. Show that if (1 02 03, < 04 C5 06 are natural numbers; then 9 | (aj Ci _ Or 9 | (aj + 4i) , for some pair G; # aj- Hint: Pigeon hole principle: Approach 1: consider the remainder of &; when divided by 9. I.e., for each i, letCi9qi + Ti,where 0 < ri < 8 If r; = Tj for some pair G; # 4j, then aj li 9(4; Yi) so 9 | (aj Wi). If not, then T: T6 are all distinct elements of the set {0,1,2,3,4,5,6,7, 8}, s0 at least 5 of these remainders are in the set {1,2,3,4,5,6,7,8}. These are your

8. Show that if (1 02 03, < 04 C5 06 are natural numbers; then 9 | (aj Ci _ Or 9 | (aj + 4i) , for some pair G; # aj- Hint: Pigeon hole principle: Approach 1: consider the remainder of &; when divided by 9. I.e., for each i, let Ci 9qi + Ti, where 0 < ri < 8 If r; = Tj for some pair G; # 4j, then aj li 9(4; Yi) so 9 | (aj Wi). If not, then T: T6 are all distinct elements of the set {0,1,2,3,4,5,6,7, 8}, s0 at least 5 of these remainders are in the set {1,2,3,4,5,6,7,8}. These are your "pigeons' DOW consider the pigeon holes {1,8}, {2,7}, {3,6}, {4,5}. Approach 2: consider the 10 natural numbers (a6 (5 ) (a6 (4) < (a6 a1) < (a6- + (1) < (a6 + (5) . Use the PHP to show that there are two distinct numbers on this list whose difference is divisible by 9. Now what?

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