Nếu n = 3k ( k ∈ N) thì 2n – 1 = 23k – 1 = 8k – 1 chia hết cho 7
Nếu n = 3k + 1 ( k ∈ N) thì 2n – 1 = 23k + 1 – 1 = 2(23k – 1) + 1 = BS 7 + 1
Nếu n = 3k + 2 ( k ∈ N) thì 2n – 1 = 23k + 2 – 1 = 4(23k – 1) + 3 = BS 7 + 3
Vậy: 2n – 1 chia hết cho 7 khi n = BS 3
\(\text{Nếu n = 3k}\)
\(\text{2^3k - 1 = 8^k - 1 = (8-1)[8^(k-1) + 8^(k-2) +..+ 8 + 1] = 7p ⋮7}\)
\(\text{Nếu n = 3k+1}\)
\(\text{2^(3k+1) -1 = 2.2^3k - 1 = 2(8^k - 1) + 1 = 2.7p + 1}\)không chia hết cho 7
\(\text{Nếu n = 3k +2}\)
\(\text{A = 2^(3k+2) -1 = 4.8^k -1 = 4(8^k - 1) + 3 = 4.7p + 3}\) không chia hết cho 7
\(\text{Vậy A = 2ⁿ -1 chia hết cho 7 }\Leftrightarrow\text{n = 3k}\left(k\in N\right)\)
Nếu n = 3k ( k ∈ N) thì 2n – 1 = 23k – 1 = 8k – 1 chia hết cho 7
Nếu n = 3k + 1 ( k ∈ N) thì 2n – 1 = 23k + 1 – 1 = 2(23k – 1) + 1 = BS 7 + 1
Nếu n = 3k + 2 ( k ∈ N) thì 2n – 1 = 23k + 2 – 1 = 4(23k – 1) + 3 = BS 7 + 3
Vậy: 2n – 1 chia hết cho 7 khi n = BS 3