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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=22, r=70)
Explanation: 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.
Correct answer: B. 3065
Explanation: 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.
Correct answer: B. 2242
Explanation: 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.
Correct answer: C. (4555=289\times15+220)
Explanation: 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.
Correct answer: B. 4
Explanation: 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.
Correct answer: C. 15
Explanation: 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.
Correct answer: C. 9
Explanation: 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.
Correct answer: A. (13q,13q+1,\ldots,13q+12)
Explanation: 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.
Correct answer: D. 0
Explanation: 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.
Correct answer: A. 0
Explanation: 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.
Correct answer: B. 124
Explanation: 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.
Correct answer: A. 39
Explanation: 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.
Correct answer: B. 44
Explanation: 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.
Correct answer: B. 13
Explanation: 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.
Correct answer: B. The remainder can be from 0 to 24
Explanation: 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.
Correct answer: A. 0
Explanation: 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.
Correct answer: C. 15
Explanation: 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.
Correct answer: B. 9
Explanation: 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.
Correct answer: A. 4
Explanation: 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.
Correct answer: A. Only 0, 1, 4 and 7
Explanation: 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.
Correct answer: A. 0
Explanation: 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.
Correct answer: C. 58
Explanation: 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.
Correct answer: A. 261
Explanation: 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.
Correct answer: C. 15
Explanation: 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.
Correct answer: A. 10
Explanation: 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.
