Question

2021 has a remainder of 5 when divided by 6, by 7, by 8, and by 9. How many positive integers, less than 2021, have this property?

Answer 1

Answer:

Rather than mechanically applying the CRT formula, the following method is usually quicker and much simpler (here so simple that it can be computed with purely mental arithmetic).

mod 35: −4≡n⟺n=−4+35j

mod 11: 2≡n=−4+35j≡−4+2j⟺6≡2j⟺j≡3⟺j=3+11k

Thus we have  n=−4+35(3+11k)=101+11⋅35k

mod 6: −1≡n≡101+11⋅35⋅k≡−1+k⟺k≡0⟺k=6m

Hence we have  n=101+11⋅35⋅6m

Step-by-step explanation:

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Answer 2

8 positive integers less than 2021 property