To find the remainder of 7^27 when divided by 1,000, we can use modular arithmetic and the concept of cyclicity.

First, let's find the remainder when 7 is raised to various powers and divided by 1,000:

7^1 ≡ 7 (mod 1,000)
7^2 ≡ 49 (mod 1,000)
7^3 ≡ 343 (mod 1,000)
7^4 ≡ 401 (mod 1,000) (since 2401 ≡ 401 (mod 1,000))
7^5 ≡ 807 (mod 1,000) (since 7 * 401 ≡ 2807 ≡ 807 (mod 1,000))
7^6 ≡ 649 (mod 1,000) (since 7 * 807 ≡ 5649 ≡ 649 (mod 1,000))
7^7 ≡ 543 (mod 1,000) (since 7 * 649 ≡ 4543 ≡ 543 (mod 1,000))
7^8 ≡ 801 (mod 1,000) (since 7 * 543 ≡ 3801 ≡ 801 (mod 1,000))
7^9 ≡ 607 (mod 1,000) (since 7 * 801 ≡ 5607 ≡ 607 (mod 1,000))
7^10 ≡ 249 (mod 1,000) (since 7 * 607 ≡ 4249 ≡ 249 (mod 1,000))
7^11 ≡ 743 (mod 1,000) (since 7 * 249 ≡ 1743 ≡ 743 (mod 1,000))
7^12 ≡ 201 (mod 1,000) (since 7 * 743 ≡ 5201 ≡ 201 (mod 1,000))
7^13 ≡ 407 (mod 1,000) (since 7 * 201 ≡ 1407 ≡ 407 (mod 1,000))
7^14 ≡ 849 (mod 1,000) (since 7 * 407 ≡ 2849 ≡ 849 (mod 1,000))
7^15 ≡ 943 (mod 1,000) (since 7 * 849 ≡ 5943 ≡ 943 (mod 1,000))
7^16 ≡ 601 (mod 1,000) (since 7 * 943 ≡ 6601 ≡ 601 (mod 1,000))
7^17 ≡ 207 (mod 1,000) (since 7 * 601 ≡ 4207 ≡ 207 (mod 1,000))
7^18 ≡ 449 (mod 1,000) (since 7 * 207 ≡ 1449 ≡ 449 (mod 1,000))
7^19 ≡ 143 (mod 1,000) (since 7 * 449 ≡ 3143 ≡ 143 (mod 1,000))
7^20 ≡ 1 (mod 1,000) (since 7 * 143 ≡ 1001 ≡ 1 (mod 1,000))

Notice that 7^20 ≡ 1 (mod 1,000), which means that every 20th power of 7 will have a remainder of 1 when divided by 1,000.

Now, we can find the remainder of 7^27 by breaking it down using the power of 20:

7^27 = (7^20) * (7^7)

Since we know that 7^20 ≡ 1 (mod 1,000), we only need to find the remainder of 7^7:

7^7 ≡ 543 (mod 1,000)

Therefore, the remainder of 7^27 when divided by 1,000 is the same as the remainder of 7^7, which is 543.
7^27을 1,000으로 나누었을 때 나머지 부분을 구하려면 모듈식 산술과 주기성 개념을 사용할 수 있습니다.

먼저 7을 다양한 거듭제곱으로 올리고 1,000으로 나눈 나머지를 구해 봅시다:

7^1 ≡ 7 (mod 1,000)
7^2 ≡ 49 (mod 1,000)
7^3 ≡ 343 (mod 1,000)
7^4 ≡ 401 (mod 1,000) (2401 ≡ 401 (mod 1,000)이므로)
7^5 ≡ 807 (mod 1,000) (7 * 401 ≡ 2807 ≡ 807 (mod 1,000)이므로)
7^6 ≡ 649 (mod 1,000) (7 * 807 ≡ 5649 ≡ 649 (mod 1,000)이므로)
7^7 ≡ 543 (mod 1,000) (7 * 649 ≡ 4543 ≡ 543 (mod 1,000)이므로)
7^8 ≡ 801 (mod 1,000) (7 * 543 ≡ 3801 ≡ 801 (mod 1,000)이므로)
7^9 ≡ 607 (mod 1,000) (7 * 801 ≡ 5607 ≡ 607 (mod 1,000)이므로)
7^10 ≡ 249 (mod 1,000) (7 * 607 ≡ 4249 ≡ 249 (mod 1,000)이므로)
7^11 ≡ 743 (mod 1,000) (7 * 249 ≡ 1743 ≡ 743 (mod 1,000)이므로)
7^12 ≡ 201 (mod 1,000) (7 * 743 ≡ 5201 ≡ 201 (mod 1,000)이므로)
7^13 ≡ 407 (mod 1,000) (7 * 201 ≡ 1407 ≡ 407 (mod 1,000)이므로)
7^14 ≡ 849 (mod 1,000) (7 * 407 ≡ 2849 ≡ 849 (mod 1,000)이므로)
7^15 ≡ 943 (mod 1,000) (7 * 849 ≡ 5943 ≡ 943 (mod 1,000)이므로)
7^16 ≡ 601 (mod 1,000) (7 * 943 ≡ 6601 ≡ 601 (mod 1,000)이므로)
7^17 ≡ 207 (mod 1,000) (7 * 601 ≡ 4207 ≡ 207 (mod 1,000)이므로)
7^18 ≡ 449 (mod 1,000) (7 * 207 ≡ 1449 ≡ 449 (mod 1,000)이므로)
7^19 ≡ 143 (mod 1,000) (7 * 449 ≡ 3143 ≡ 143 (mod 1,000)이므로)
7^20 ≡ 1 (mod 1,000) (7 * 143 ≡ 1001 ≡ 1 (mod 1,000)이므로)

7^20 ≡ 1 (mod 1,000)은 7의 20번째 거듭제곱을 1,000으로 나누었을 때 나머지 부분이 1이 된다는 것을 의미합니다.

이제 20의 거듭제곱을 사용하여 7^27의 나머지를 구할 수 있습니다:

7^27 = (7^20) * (7^7)

7^20 ≡ 1 (mod 1,000)이라는 것을 알았으므로 7^7의 나머지 부분만 구하면 됩니다:

7^7 ≡ 543 (mod 1,000)

따라서 7^27을 1,000으로 나눈 나머지는 7^7의 나머지인 543과 동일합니다.