Given the number \( N \), which is a multiple of \( 7 \), has a base \( 2 \) representation of \( 10011010011ABC110_2 \). To determine the values of \( A, B, \) and \( C \), we convert this binary number into its decimal form and then ensure that the resulting number is divisible by \( 7 \).
First, express \( N \) in decimal terms:
\[
N = 1 \cdot 2^{16} + 0 \cdot 2^{15} + 0 \cdot 2^{14} + 1 \cdot 2^{13} + 1 \cdot 2^{12} + 0 \cdot 2^{11} + 1 \cdot 2^{10} + 0 \cdot 2^9 + 0 \cdot 2^8 + 1 \cdot 2^7 + A \cdot 2^6 + B \cdot 2^5 + C \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0
\]
Calculate each power of 2:
\[
2^{16} = 65536, \quad 2^{13} = 8192, \quad 2^{12} = 4096, \quad 2^{10} = 1024, \quad 2^7 = 128, \quad 2^6 = 64, \quad 2^5 = 32, \quad 2^4 = 16, \quad 2^3 = 8, \quad 2^2 = 4, \quad 2^1 = 2, \quad 2^0 = 1
\]
Substitute back into the expression for \( N \):
\[
N = 65536 + 8192 + 4096 + 1024 + 128 + A \cdot 64 + B \cdot 32 + C \cdot 16 + 8 + 4 + 2
\]
\[
N = 65536 + 8192 + 4096 + 1024 + 128 + 14 + A \cdot 64 + B \cdot 32 + C \cdot 16
\]
\[
N = 78990 + 64A + 32B + 16C
\]
Given \( N \) is divisible by 7, verifying the expression modulo 7:
\[
N \equiv 78990 + 64A + 32B + 16C \pmod{7}
\]
Simplify each term modulo 7:
\[
78990 \equiv 2 \pmod{7} \quad \text{(since } 78990 = 7 \times 11284 + 2\text{)}
\]
\[
64 \equiv 1 \pmod{7} \quad (64 = 7 \times 9 + 1)
\]
\[
32 \equiv 4 \pmod{7} \quad (32 = 7 \times 4 + 4)
\]
\[
16 \equiv 2 \pmod{7} \quad (16 = 7 \times 2 + 2)
\]
Thus:
\[
N \equiv 2 + A + 4B + 2C \pmod{7}
\]
To be a multiple of 7:
\[
2 + A + 4B + 2C \equiv 0 \pmod{7}
\]
\[
A + 4B + 2C \equiv -2 \equiv 5 \pmod{7}
\]
We solve for non-negative integer solutions \(A\), \(B\), and \(C\) between 0 and 1 (binary digits). Evaluating the possible values systematically:
We find the values \( A = 1, B = 1, C = 0 \) satisfy the equation:
\[
1 + 4 \cdot 1 + 2 \cdot 0 = 1 + 4 = 5 \equiv 5 \pmod{7}
\]
The ordered triple \((A, B, C)\) is:
\[
\boxed{(1, 1, 0)}
\]
The number $N$ is a multiple of $7$. The base $2$ representation of $N$ is
10011010011ABC110_2.
Compute the ordered triple of digits $(A,B,C)$.
1 answer