Euclid’s Division Lemma

Step 1: The remainder condition is (0\le r<125). Step 2: The greatest integer smaller than 125 is 124. Step 3: The remainder is never equal to the divisor.

Step 1: For seven times the number, the remainder part is (7\times46=322). Step 2: (322=47\times6+40), so the remainder should be 40. Step 3: After multiplication, always reduce the result below the divisor.

Step 1: Find the nearest lower multiple of 221 below 6895. Step 2: (221\times31=6851), so the remainder is (6895-6851=44). Step 3: With large numbers, the nearest lower multiple method saves time.

Step 1: For the cube, check (7^3=343). Step 2: (343=15\times22+13), so the remainder is 13. Step 3: In power questions, reduce remainders along the way to keep calculations short.

Step 1: In Euclid’s lemma, (0\le r<b). Step 2: Here (b=25), so the remainder can be from 0 to 24. Step 3: Include 0 in the list of remainders and do not include 25.

Step 1: The remainder of (p) is 16. Step 2: The remainder of (6p-20) comes from (6\times16-20=76). Step 3: Since (76=19\times4), the final remainder is 0.

Step 1: The two remainders are 18 and 20. Step 2: The sum remainder comes from (18+20=38). Step 3: Since (38=23+15), the final remainder is 15.

Step 1: For multiplication, multiply the remainders 24 and 27. Step 2: (24\times27=648), and (648=29\times22+10). Step 3: So the correct remainder is 10; the full product is not the answer.

Step 1: For the cube, check (5^3=125). Step 2: (125=11\times11+4), so the remainder is 4. Step 3: In powers, working with small remainders avoids large calculations.

Step 1: On division by 9, remainders can be from 0 to 8. Step 2: Their square remainders are 0, 1, 4, 0, 7, 7, 0, 4, and 1. Step 3: So the square remainder can only be 0, 1, 4, or 7.

Step 1: The remainder of (N) is 33. Step 2: 90 leaves remainder 8 when divided by 41. Step 3: The total remainder is (33+8=41), so the final remainder is 0.

Step 1: When the divisor is 58, the remainder can be from 0 to 57. Step 2: 58 is equal to the divisor, so it cannot be a remainder. Step 3: In definition-based questions, check the condition (r<b) first.

Step 1: (307\times23=7061). Step 2: (7341-7061=280), so the remainder is 280. Step 3: Choose a multiple that does not exceed the number and leaves a difference smaller than the divisor.

Step 1: The remainder of (t) is 27. Step 2: The remainder part of (5t+16) is (5\times27+16=151). Step 3: (151=34\times4+15), so the final remainder is 15.

Step 1: For eleven times the number, the remainder part is (11\times12=132). Step 2: (132=61\times2+10), so the final remainder is 10. Step 3: After multiplication, reduce the result by the divisor to make a valid remainder.

Step 1: On division by 9, possible remainders are from 0 to 8. Step 2: Nine consecutive integers cover all these remainders once. Step 3: The number with remainder 0 is divisible by 9.

Step 1: The remainders in the two forms are 3 and 7. Step 2: (3^2=9) and (7^2=49), and both leave remainder 9 when divided by 10. Step 3: In form-based questions, work only with the remainder.

Step 1: (a-119=67q+52-119=67q-67). Step 2: This can be written as (67(q-1)+0). Step 3: The number is exactly divisible by 67, so the remainder is 0.

Step 1: For the square, take (21^2=441). Step 2: (441=26\times16+25), so the remainder should be 25. Step 3: The final answer must always be less than 26.

Step 1: The remainder of (x) is 11. Step 2: The remainder of (x^2+x) comes from (11^2+11=132). Step 3: Since 132 is exactly divisible by 12, the remainder is 0.