Euclid’s Division Lemma

Step 1: Check the squares of the given remainders. Step 2: (1^2) and (9^2) leave remainder 1, while (3^2) and (7^2) leave remainder 9. Step 3: When listing possible remainders, count repeated results only once.

Step 1: For eleven times the number, the remainder part is (11\times8=88). Step 2: (88=39\times2+10), so the final remainder is 10. Step 3: After multiplication, divide the result again by the divisor.

Step 1: The remainder of (a) is 7. Step 2: The remainder of (a^2+5) comes from (7^2+5=54). Step 3: (54=12\times4+6), so the final remainder is 6.

Step 1: In (4a+3b), the remainder part is (4\times9+3\times13=75). Step 2: (75=17\times4+7), so the remainder is 7. Step 3: In multi-term expressions, handle each term’s remainder separately.

Step 1: The original remainder is 57. Step 2: 175 leaves remainder 1 when divided by 58. Step 3: (57+1=58), so the final remainder is 0.

Step 1: Replace (n) by its remainder 4. Step 2: The remainder of (n^2+n+1) comes from (4^2+4+1=21). Step 3: Dividing 21 by 5 gives remainder 1.

Step 1: The remainder of (a) is 31. Step 2: For (8a), check (8\times31=248). Step 3: (248=42\times5+38), so the remainder is 38.

Step 1: The remainder must satisfy (0\le r<97). Step 2: 97 is equal to the divisor, so it cannot be a valid remainder. Step 3: In definition questions, check the remainder range first.

Step 1: (n^2+2n+1=(n+1)^2). Step 2: Since the remainder of (n) is 13, the remainder of (n+1) is 0. Step 3: When (n+1) is divisible by 14, its square is also divisible by 14.

Step 1: Add the remainders: (29+31+6=66). Step 2: 66 is exactly divisible by 33. Step 3: Therefore, the final remainder is 0; adding only the remainders is a quick method for multi-term expressions.

Step 1: Find the nearest lower multiple of 173 below 3876. Step 2: (173\times22=3806), so the remainder is (3876-3806=70). Step 3: In a valid answer, the remainder must be less than 173.

Step 1: The number has the form (73\times41+r), where (0\le r<73). Step 2: The greatest remainder is 72, so the number is (2993+72=3065). Step 3: For the greatest number, take the remainder one less than the divisor.

Step 1: The number is (118\times19+r). Step 2: For the least value, take remainder 0, so the number is (118\times19=2242). Step 3: For least value questions, using remainder zero is the most direct method.

Step 1: In standard form, the remainder must be from 0 to 288. Step 2: (289\times15=4335), so (4555=4335+220). Step 3: A negative remainder or a remainder greater than 289 does not make the correct Euclidean form.

Step 1: The remainder of (x) is 49. Step 2: 111 leaves remainder 7 when divided by 52, so the total remainder is (49+7=56). Step 3: Since (56=52+4), the final remainder is 4.

Step 1: Replace (n) by its remainder 23. Step 2: The remainder of (8n+17) comes from (8\times23+17=201). Step 3: (201=31\times6+15), so the final remainder is 15.

Step 1: The square remainder comes from dividing (14^2=196) by 17. Step 2: (196=17\times11+9), so the remainder is 9. Step 3: In square questions, squaring the remainder is faster than using the whole number.

Step 1: On division by 13, remainders can be from 0 to 12. Step 2: Therefore, all forms are from (13q) to (13q+12). Step 3: Include zero remainder, but do not include 13.

Step 1: The remainder of (a) is 82, one less than 83. Step 2: Adding 1 gives (83q+83=83(q+1)). Step 3: The number becomes exactly divisible by 83, so the remainder is 0.

Step 1: Write (m=43q+9). Step 2: (m-95=43q+9-95=43q-86=43(q-2)). Step 3: It is exactly divisible by 43, so the remainder is 0.