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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=20, r=201)
Explanation: Step 1: Find the nearest lower multiple of 238 below 4961. Step 2: (238\times20=4760), so the remainder is (4961-4760=201). Step 3: Since the remainder is smaller than 238, this is the valid Euclidean form.
Correct answer: B. 4088
Explanation: Step 1: The number is of the form (87\times46+r), where (0\le r<87). Step 2: The greatest remainder is 86, so the number is (4002+86=4088). Step 3: For the greatest value, the remainder is always one less than the divisor.
Correct answer: A. 3289
Explanation: Step 1: The number is (143\times23+r). Step 2: For the least value, the remainder is 0, so the number is (143\times23=3289). Step 3: In least-value questions, taking the remainder as zero is the clearest method.
Correct answer: C. (6127=391\times15+262)
Explanation: Step 1: In standard form, the remainder must be from 0 to 390. Step 2: (391\times15=5865), so (6127=5865+262). Step 3: A negative remainder or a remainder greater than 391 does not make the correct Euclidean form.
Correct answer: B. 3
Explanation: Step 1: The remainder of (x) is 64. Step 2: 140 leaves remainder 6 when divided by 67, so the total remainder is (64+6=70). Step 3: Since (70=67+3), the final remainder is 3.
Correct answer: B. 1
Explanation: Step 1: Replace (n) by its remainder 31. Step 2: The remainder of (9n+22) comes from (9\times31+22=301). Step 3: Since (301=43\times7+0), the remainder is 0.
Correct answer: A. 9
Explanation: Step 1: The square remainder comes from dividing (16^2=256) by 19. Step 2: (256=19\times13+9), so the remainder is 9. Step 3: In square questions, square only the remainder instead of the full number.
Correct answer: A. (17q,17q+1,\ldots,17q+16)
Explanation: Step 1: On division by 17, remainders can be from 0 to 16. Step 2: Therefore, all forms are from (17q) to (17q+16). Step 3: Include zero remainder, but do not include 17.
Correct answer: D. 0
Explanation: Step 1: The remainder of (a) is 96, one less than 97. Step 2: Adding 1 gives (97q+97=97(q+1)). Step 3: The number becomes exactly divisible by 97, so the remainder is 0.
Correct answer: A. 0
Explanation: Step 1: Write (m=53q+12). Step 2: (m-118=53q+12-118=53q-106=53(q-2)). Step 3: It is exactly divisible by 53, so the remainder is 0.
Correct answer: B. 143
Explanation: Step 1: The remainder condition is (0\le r<144). Step 2: The greatest integer smaller than 144 is 143. Step 3: The remainder is never equal to the divisor.
Correct answer: A. 49
Explanation: Step 1: For nine times the number, the remainder part is (9\times58=522). Step 2: (522=59\times8+50), so the remainder is 50. Step 3: After multiplication, always reduce the result below the divisor.
Correct answer: A. 289
Explanation: Step 1: Find the nearest lower multiple of 359 below 9264. Step 2: (359\times25=8975), so the remainder is (9264-8975=289). Step 3: With large numbers, the nearest lower multiple method saves time.
Correct answer: C. 17
Explanation: Step 1: For the cube, check (11^3=1331). Step 2: (1331=18\times73+17), so the remainder is 17. Step 3: In power questions, reduce remainders repeatedly to keep the calculation simple.
Correct answer: B. The remainder can be from 0 to 31
Explanation: Step 1: In Euclid’s lemma, (0\le r<b). Step 2: Here (b=32), so the remainder can be from 0 to 31. Step 3: Include 0 in the list of remainders and do not include 32.
Correct answer: A. 0
Explanation: Step 1: The remainder of (p) is 19. Step 2: The remainder of (7p-18) comes from (7\times19-18=115). Step 3: Since (115=23\times5), the final remainder is 0.
Correct answer: C. 22
Explanation: Step 1: The two remainders are 25 and 28. Step 2: The sum remainder comes from (25+28=53). Step 3: Since (53=31+22), the final remainder is 22.
Correct answer: B. 10
Explanation: Step 1: For multiplication, multiply the remainders 32 and 35. Step 2: (32\times35=1120), and (1120=37\times30+10). Step 3: In product questions, the answer is the final remainder, not the full product.
Correct answer: B. 8
Explanation: Step 1: For the cube, check (6^3=216). Step 2: (216=13\times16+8), so the remainder is 8. Step 3: In powers, working with small remainders avoids large calculations.
Correct answer: A. Only 0, 1, 3, 4, 5 and 9
Explanation: Step 1: On division by 11, remainders can be from 0 to 10. Step 2: Their distinct square remainders are 0, 1, 3, 4, 5, and 9. Step 3: In square-remainder questions, making a short list of possible remainders is useful.
Correct answer: A. 0
Explanation: Step 1: The remainder of (N) is 39. Step 2: 102 leaves remainder 8 when divided by 47. Step 3: The total remainder is (39+8=47), so the final remainder is 0.
Correct answer: C. 69
Explanation: Step 1: When the divisor is 69, the remainder can be from 0 to 68. Step 2: 69 is equal to the divisor, so it cannot be a remainder. Step 3: In definition-based questions, check the condition (r<b) first.
Correct answer: A. 0
Explanation: Step 1: (412\times23=9476) and (412\times24=9888). Step 2: The nearest lower multiple below 9876 is 9476, so the remainder is (400). Step 3: Since the remainder is less than 412, it is valid.
Correct answer: B. 9
Explanation: Step 1: The remainder of (t) is 37. Step 2: The remainder part of (6t+17) is (6\times37+17=239). Step 3: (239=46\times5+9), so the final remainder is 9.
Correct answer: B. 49
Explanation: Step 1: For thirteen times the number, the remainder part is (13\times15=195). Step 2: (195=73\times2+49), so the final remainder is 49. Step 3: After multiplication, reduce the result by the divisor to make a valid remainder.
