Finding Remainder of Product Division by 27

The question asks for the remainder when the product $127 \times 141 \times 147 \times 162 \times 177$ is divided by 27.

We can use the properties of modular arithmetic. The remainder of a product of numbers divided by a divisor is the same as the remainder of the product of their individual remainders divided by that same divisor.

Mathematically, if we want to find $(a \times b \times c \times d \times e) \pmod{n}$, we can first find the remainders of $a, b, c, d, e$ when divided by $n$, let them be $r_a, r_b, r_c, r_d, r_e$. Then, the required remainder is $(r_a \times r_b \times r_c \times r_d \times r_e) \pmod{n}$.

Step 1: Calculate Individual Remainders

Let's find the remainder of each number when divided by 27:

• $127 \div 27$: $127 = 4 \times 27 + 19$. So, $127 \equiv 19 \pmod{27}$.
• $141 \div 27$: $141 = 5 \times 27 + 6$. So, $141 \equiv 6 \pmod{27}$.
• $147 \div 27$: $147 = 5 \times 27 + 12$. So, $147 \equiv 12 \pmod{27}$.
• $162 \div 27$: $162 = 6 \times 27 + 0$. So, $162 \equiv 0 \pmod{27}$.
• $177 \div 27$: $177 = 6 \times 27 + 15$. So, $177 \equiv 15 \pmod{27}$.

Step 2: Calculate Remainder of the Product

Now, we multiply these remainders and find the remainder when divided by 27:

Remainder = $(19 \times 6 \times 12 \times 0 \times 15) \pmod{27}$

Since one of the factors in the product of remainders is 0 (from the number 162), the entire product becomes 0.

Remainder = $0 \pmod{27}$

Remainder = $0$

Conclusion

The remainder when $127 \times 141 \times 147 \times 162 \times 177$ is divided by 27 is 0.