Consecutive Remainders
Find the sum of all x ∈ [ 0 , 5 0 ] such that at least one of the following conditions is satisfied:
- x ≡ 2 ( m o d 3 ) ,
- x ≡ 3 ( m o d 4 ) , or
- x ≡ 4 ( m o d 5 ) .
The answer is 746.
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1 solution
I approached in this way. Each congruence is − 1 . Thus, the number x is congruent to − 1 (mod 3 × 4 × 5 . Thus, the smallest natural number for x is 5 9 , which is not in the range.
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just use Ms.Excel
The word 'or' in the question
Won't CRT work? @Finn Hulse - Or only casework will?
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I did it by CRT . Then I added the three APs and then substracted the conditions for which any two of the given conditions were satisfied
If x satisfies all the congruences, it is either too big or too small.
Our answer is this, which is very simple to calculate using the 1 + ⋯ + n = 2 n ( n + 1 ) formula.
i = 1 ∑ 1 7 ( 3 i − 1 ) + i = 1 ∑ 1 2 ( 4 i − 1 ) + i = 1 ∑ 1 0 ( 5 i − 1 ) − i = 1 ∑ 4 ( 1 2 i − 1 ) − i = 1 ∑ 4 ( 2 0 i − 1 ) − i = 1 ∑ 3 ( 1 5 i − 1 )
Here's what I did. I made a list of all values n such that n ≡ 2 ( m o d 3 ) and the like. Then, I crossed out each term that appeared on each list more than once. From here, I just added, producing the number 7 4 6 .