Let's solve this problem step-by-step.
We are given:
- [tex]\( n \)[/tex] leaves a remainder of 3 when divided by 5.
- [tex]\( k \)[/tex] leaves a remainder of 2 when divided by 5.
This can be written mathematically as:
[tex]\[ n \equiv 3 \pmod{5} \][/tex]
[tex]\[ k \equiv 2 \pmod{5} \][/tex]
We need to find the remainder when the product [tex]\( nk \)[/tex] is divided by 5.
First, express [tex]\( n \)[/tex] and [tex]\( k \)[/tex] in terms of their remainders when divided by 5:
[tex]\[ n = 5a + 3 \][/tex]
[tex]\[ k = 5b + 2 \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are integers.
Now, consider the product [tex]\( nk \)[/tex]:
[tex]\[ nk = (5a + 3)(5b + 2) \][/tex]
Expand the product using distribution:
[tex]\[ nk = 5a \cdot 5b + 5a \cdot 2 + 3 \cdot 5b + 3 \cdot 2 \][/tex]
Simplify the expression:
[tex]\[ nk = 25ab + 10a + 15b + 6 \][/tex]
Notice that every term except the constant term contains a factor of 5:
[tex]\[ 25ab \text{ is divisible by 5} \][/tex]
[tex]\[ 10a \text{ is divisible by 5} \][/tex]
[tex]\[ 15b \text{ is divisible by 5} \][/tex]
Therefore, these terms will not affect the remainder when divided by 5. The only term that affects the remainder is the constant term 6.
We now need to find the remainder of 6 when divided by 5:
[tex]\[ 6 \div 5 = 1 \text{ remainder } 1 \][/tex]
Thus, the remainder when the product [tex]\( nk \)[/tex] is divided by 5 is:
[tex]\[ \boxed{1} \][/tex]
