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SubjectsMathematics
यूक्लिड का विभाजन प्रमेय
In Class 10 Mathematics, under the chapter Real Numbers, Euclid’s Division Lemma introduces the relationship a = bq + r, where a and b are positive integers, q is the quotient, and the remainder r satisfies 0 ≤ r < b. Students learn how repeated division forms Euclid’s division algorithm and use it to find the highest common factor (HCF) of two numbers. The topic also strengthens understanding of divisibility, quotients, remainders, and the logical steps used in number-theory proofs.
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.
