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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=16, r=51)
Explanation: Step 1: Find the nearest lower multiple of 156 below 2547. Step 2: (156\times16=2496), so the remainder is (2547-2496=51). Step 3: In a valid Euclidean form, the remainder must be smaller than the divisor.
Correct answer: B. 2431
Explanation: Step 1: The number is of the form (64\times37+r), where (0\le r<64). Step 2: The greatest remainder is 63, so the number is (2368+63=2431). Step 3: For the greatest value, take the remainder as one less than the divisor.
Correct answer: A. 2548
Explanation: Step 1: The number is (91\times28+r). Step 2: For the least value, the remainder is 0, so the number is (91\times28=2548). Step 3: In minimum value questions, taking remainder zero gives the answer quickly.
Correct answer: A. (3199=247\times12+235)
Explanation: Step 1: In Euclidean form, the remainder is non-negative and smaller than 247. Step 2: (247\times12=2964), so (3199=2964+235). Step 3: A negative remainder or a remainder bigger than the divisor is not the standard form.
Correct answer: B. 1
Explanation: Step 1: The remainder of (x) is 41. Step 2: 89 leaves remainder 3 when divided by 43, so the total remainder is (41+3=44). Step 3: Since (44=43+1), the final remainder is 1.
Correct answer: A. 13
Explanation: Step 1: Replace (n) by its remainder 17. Step 2: The remainder of (7n+13) comes from (7\times17+13=132). Step 3: (132=29\times4+16), so the correct remainder is 16.
Correct answer: A. 4
Explanation: Step 1: The square remainder comes from dividing (11^2=121) by 13. Step 2: (121=13\times9+4), so the remainder is 4. Step 3: In square questions, squaring only the remainder is faster than using the whole number.
Correct answer: A. (11q,11q+1,\ldots,11q+10)
Explanation: Step 1: On division by 11, remainders can be from 0 to 10. Step 2: Therefore, all forms are from (11q) to (11q+10). Step 3: Include remainder 0, but do not include 11.
Correct answer: D. 0
Explanation: Step 1: The remainder of (a) is 70, one less than 71. Step 2: Adding 1 gives (71q+71=71(q+1)). Step 3: The number becomes exactly divisible by 71, so the remainder is 0.
Correct answer: C. 29
Explanation: Step 1: Write (m=31q+6). Step 2: (m-68=31q-62=31(q-2)), so the remainder is 0. Step 3: In subtraction, check the final form as divisor times quotient plus remainder.
Correct answer: B. 107
Explanation: Step 1: The remainder condition is (0\le r<108). Step 2: The greatest integer smaller than 108 is 107. Step 3: Remember in exams that the remainder is never equal to the divisor.
Correct answer: C. 29
Explanation: Step 1: For five times the number, the remainder part is (5\times33=165). Step 2: (165=34\times4+29), so the final remainder is 29. Step 3: After multiplication, always reduce the result below the divisor.
Correct answer: B. 121
Explanation: Step 1: Find the nearest lower multiple of 132 below 4217. Step 2: (132\times31=4092), so the remainder is (4217-4092=125). Step 3: For large numbers, the nearest lower multiple method is useful.
Correct answer: B. 5
Explanation: Step 1: For the cube, consider (5^3=125). Step 2: (125=12\times10+5), so the remainder is 5. Step 3: In power questions, reduce remainders along the way to keep calculation easy.
Correct answer: B. The remainder can be from 0 to 17
Explanation: Step 1: In Euclid’s lemma, (0\le r<b). Step 2: Here (b=18), so the remainder can be from 0 to 17. Step 3: Include 0 in the list of remainders and do not include the divisor.
Correct answer: A. 0
Explanation: Step 1: The remainder of (p) is 14. Step 2: The remainder of (5p-19) comes from (5\times14-19=51). Step 3: Since (51=17\times3), the final remainder is 0.
Correct answer: C. 12
Explanation: Step 1: The two remainders are 16 and 15. Step 2: The sum remainder comes from (16+15=31). Step 3: Since (31=19+12), the final remainder is 12.
Correct answer: B. 8
Explanation: Step 1: For multiplication, multiply the remainders 19 and 21. Step 2: (19\times21=399), and (399=23\times17+8). Step 3: In product questions, the final answer must be the remainder, not the full product.
Correct answer: B. 1
Explanation: Step 1: For the cube, consider (4^3=64). Step 2: (64=9\times7+1), so the remainder is 1. Step 3: In powers, working with the small remainder avoids large calculations.
Correct answer: A. Only 0, 1, 2 and 4
Explanation: Step 1: On division by 7, remainders can be from 0 to 6. Step 2: Their square remainders are 0, 1, 4, 2, 2, 4, and 1. Step 3: So the square remainder can only be 0, 1, 2, or 4.
Correct answer: A. 0
Explanation: Step 1: The remainder of (N) is 30. Step 2: 81 leaves remainder 7 when divided by 37. Step 3: The total remainder is (30+7=37), so the final remainder is 0.
Correct answer: C. 46
Explanation: Step 1: When the divisor is 46, the remainder can be from 0 to 45. Step 2: 46 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: C. 14
Explanation: Step 1: (211\times25=5275). Step 2: (5289-5275=14), so the remainder is 14. Step 3: While dividing, choose a multiple that does not exceed the given number.
Correct answer: A. 9
Explanation: Step 1: The remainder of (t) is 19. Step 2: The remainder part of (4t+11) is (4\times19+11=87). Step 3: (87=26\times3+9), so the final remainder is 9.
Correct answer: C. 29
Explanation: Step 1: For nine times the number, the remainder part is (9\times9=81). Step 2: (81=52+29), so the final remainder is 29. Step 3: After multiplication, reduce the result by the divisor to make a valid remainder.
