Euclid’s Division Lemma

Correct answer: A. (q=12, r=21)

Explanation: Step 1: Find the nearest lower multiple of 112 below 1365. Step 2: (112\times12=1344), so the remainder is (1365-1344=21). Step 3: In exams, always check that the final remainder is smaller than the divisor.

Correct answer: B. 896

Explanation: Step 1: The number has the form (39\times22+r), where (0\le r<39). Step 2: The greatest remainder is 38, so the number is (858+38=896). Step 3: For the greatest number, take the remainder one less than the divisor.

Correct answer: B. 1026

Explanation: Step 1: The number is (57\times18+r). Step 2: For the least value, (r=0), so the number is (57\times18=1026). Step 3: For a minimum value, start with remainder zero.

Correct answer: A. (947=73\times12+71)

Explanation: Step 1: A valid remainder must be from 0 to 72. Step 2: (73\times12=876), so (947=876+71), and 71 is valid. Step 3: A form with a negative remainder may look close, but it is not Euclidean form.

Correct answer: B. 1

Explanation: Step 1: The remainder of (x) is 25. Step 2: Adding 28 gives total remainder (25+28=53), and (53=26\times2+1). Step 3: After addition, reduce the remainder again by the divisor.

Correct answer: A. 9

Explanation: Step 1: Let (n=19q+8). Step 2: The remainder of (5n+7) comes from (5\times8+7=47), and (47=19\times2+9). Step 3: In a linear expression, using the remainder keeps the calculation short.

Correct answer: A. 4

Explanation: Step 1: The square remainder comes from dividing (9^2=81) by 11. Step 2: (81=11\times7+4), so the remainder is 4. Step 3: In square questions, square the remainder instead of the whole number.

Correct answer: A. (9q,9q+1,\ldots,9q+8)

Explanation: Step 1: On division by 9, possible remainders are from 0 to 8. Step 2: Therefore, all forms are from (9q) to (9q+8). Step 3: Include remainder 0 and do not include remainder 9.

Correct answer: D. 0

Explanation: Step 1: The remainder of (a) is 30, one less than 31. Step 2: Adding 1 gives (31q+31=31(q+1)), so the remainder is 0. Step 3: Adding 1 to a remainder one less than the divisor gives the next exact multiple.

Correct answer: A. 15

Explanation: Step 1: Write (m=17q+4). Step 2: (m-23=17q-19=17(q-2)+15), so the remainder is 15. Step 3: If subtraction gives a negative remainder, add the divisor as needed.

Correct answer: B. 51

Explanation: Step 1: The remainder condition is (0\le r<52). Step 2: The greatest integer smaller than 52 is 51, so it is the greatest possible remainder. Step 3: A remainder can never be equal to the divisor.

Correct answer: A. 25

Explanation: Step 1: For three times the number, the remainder part is (3\times27=81). Step 2: (81=28\times2+25), so the final remainder is 25. Step 3: After multiplication, reduce the remainder below the divisor.

Correct answer: B. 14

Explanation: Step 1: Find the nearest lower multiple of 41 below 875. Step 2: (41\times21=861), so the remainder is (875-861=14). Step 3: The nearest lower multiple method saves time with larger numbers.

Correct answer: B. 7

Explanation: Step 1: For the cube, take (3^3=27). Step 2: Dividing 27 by 10 gives remainder 7. Step 3: In power questions, use the smaller remainder instead of the full number.

Correct answer: B. The remainder can be from 0 to 13

Explanation: Step 1: In Euclid’s lemma, (0\le r<b). Step 2: Here (b=14), so the remainder can be from 0 to 13. Step 3: Include 0 in the list of remainders and exclude the divisor itself.

Correct answer: A. 9

Explanation: Step 1: The remainder of (p) is 11. Step 2: The remainder of (4p-9) comes from (4\times11-9=35), and (35=13\times2+9). Step 3: Always reduce the final remainder below the divisor.

Correct answer: B. 5

Explanation: Step 1: The two remainders are 6 and 7. Step 2: The sum remainder comes from (6+7=13), and (13=8+5). Step 3: If the sum of remainders is greater than the divisor, reduce it again.

Correct answer: B. 10

Explanation: Step 1: For multiplication, multiply the remainders 12 and 15. Step 2: (12\times15=180), and (180=17\times10+10). Step 3: In product questions, multiply the remainders and then find the final remainder.

Correct answer: B. 6

Explanation: Step 1: For the cube, consider (5^3=125). Step 2: (125=7\times17+6), so the remainder is 6. Step 3: In higher powers, reduce remainders along the way to keep calculation easy.

Correct answer: A. Only 0, 1 and 4

Explanation: Step 1: The possible remainders of a number are 0, 1, 2, 3, and 4. Step 2: Their square remainders are 0, 1, 4, 4, and 1 respectively. Step 3: A square divided by 5 never leaves remainder 2 or 3.

Correct answer: C. 2

Explanation: Step 1: The remainder of (N) is 17. Step 2: (49) leaves remainder 5 on division by 22, so (17+5=22), which gives remainder 0. Step 3: After reducing the added number, reduce the final sum again by the divisor.

Correct answer: C. 24

Explanation: Step 1: When the divisor is 24, the remainder can be from 0 to 23. Step 2: 24 is equal to the divisor, so it cannot be a remainder. Step 3: In remainder questions, carefully check any option equal to the divisor.

Correct answer: C. 7

Explanation: Step 1: (67\times25=1675). Step 2: (1682-1675=7), so the remainder is 7. Step 3: While dividing, choose a multiple that does not exceed the given number.

Correct answer: B. 7

Explanation: Step 1: The remainder of (t) is 13. Step 2: The remainder part of (3t+4) is (3\times13+4=43), and (43=18\times2+7). Step 3: The final remainder must be reduced below 18.

Correct answer: B. 10

Explanation: Step 1: For seven times the number, the remainder part is (7\times6=42). Step 2: (42=32+10), so the final remainder is 10. Step 3: After multiplication, reduce the result again by the divisor.