To solve for [tex]\( Y \)[/tex] given the following congruences:
[tex]\[ A \bmod 17 = 15 \][/tex]
[tex]\[ B \bmod 17 = 9 \][/tex]
and knowing that:
[tex]\[ (A \cdot B) \bmod 17 = Y \][/tex]
we can proceed with the calculations as follows:
1. Identify the given values:
- [tex]\( A \equiv 15 \pmod{17} \)[/tex]
- [tex]\( B \equiv 9 \pmod{17} \)[/tex]
2. Multiply the two congruence values:
- Calculate [tex]\( A \cdot B \)[/tex].
[tex]\[ A \cdot B = 15 \cdot 9 \][/tex]
3. Compute the product:
[tex]\[ 15 \cdot 9 = 135 \][/tex]
4. Find the modulo of the product with 17:
- To find [tex]\( Y \)[/tex]:
[tex]\[ 135 \bmod 17 \][/tex]
5. Perform the division to find the remainder:
- Divide 135 by 17:
[tex]\[ 135 \div 17 \approx 7.94 \][/tex]
- Determine the integer part of the division (which is 7), and use it to find the remainder:
[tex]\[ 135 - 17 \times 7 = 135 - 119 = 16 \][/tex]
Thus, the remainder when 135 is divided by 17, and therefore the value of [tex]\( Y \)[/tex], is:
[tex]\[ Y = 16 \][/tex]
Therefore, [tex]\( (A \cdot B) \bmod 17 = 16 \)[/tex].
