Question

\(a\in N\) . Biết a chia 6 dư 1 , chỉ 7 dư 3 . Hội chia 42 dư bao nhiêu ?

Answer 1

dư 31

Answer 2

Ta có: \(a=6m+1=7n+3\) \(\left(m,n\in N\right)\)

\(\Rightarrow a+11=6m+1+11=7n+3+11\)

\(\Rightarrow a+11=6m+12=7n+14\)

\(\Rightarrow a+11=6.\left(m+2\right)=7.\left(n+2\right)\)

\(\Rightarrow a+11\in BC\left(6;7\right)\)

Vì \(\left(6;7\right)=1\Rightarrow BCNN\left(6;7\right)=6.7=42\)

\(\Rightarrow a+11\in B\left(42\right)\)

\(\Rightarrow a+11=42k\left(k\in N\right)\)

\(\Rightarrow a=42k-11\)

\(\Rightarrow a=42k-42+31\)

\(\Rightarrow a=42\left(k-1\right)+31\)

Vậy a chia cho 42 dư 31.