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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=17)
Explanation: Step 1: Find the nearest lower multiple of 84 below 1025. Step 2: (84\times12=1008), so the remainder is (1025-1008=17). Step 3: The final remainder must be less than 84 for the form to be valid.
Correct answer: B. 867
Explanation: Step 1: The number is of the form (31\times27+r), where (0\le r<31). Step 2: The greatest remainder is 30, so the number is (837+30=867). Step 3: For the greatest value, take the remainder as one less than the divisor.
Correct answer: B. 874
Explanation: Step 1: The number is (46\times19+r). Step 2: For the least value, take remainder 0, so the number is (46\times19=874). Step 3: For minimum value questions, taking remainder zero is the safest method.
Correct answer: A. (703=58\times12+7)
Explanation: Step 1: A valid remainder must be between 0 and 57. Step 2: (58\times12=696), so (703=696+7), and 7 is valid. Step 3: Along with calculation, also check the range of the remainder.
Correct answer: A. 0
Explanation: Step 1: The remainder of (x) is 14. Step 2: Adding 16 gives total remainder (14+16=30), which is exactly divisible by 15. Step 3: After addition, reduce the remainder again by the divisor.
Correct answer: B. 12
Explanation: Step 1: Let (n=17q+6). Step 2: (4n+5=68q+24+5=68q+29=17(4q+1)+12). Step 3: In a linear expression, first multiply the remainder and then reduce by the divisor.
Correct answer: B. 4
Explanation: Step 1: The square remainder comes from dividing (7^2=49) by 9. Step 2: (49=9\times5+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. (6q,6q+1,6q+2,6q+3,6q+4,6q+5)
Explanation: Step 1: On division by 6, possible remainders are 0, 1, 2, 3, 4, and 5. Step 2: So the forms are from (6q) to (6q+5). Step 3: Include remainder 0 and do not include 6 in the complete list.
Correct answer: C. 0
Explanation: Step 1: The remainder of (a) is 22, one less than 23. Step 2: Adding 1 gives (23q+23=23(q+1)), so the remainder is 0. Step 3: Adding 1 to a remainder one less than the divisor takes the number to the next multiple.
Correct answer: A. 10
Explanation: Step 1: Write (m=12q+5). Step 2: (m-19=12q-14=12(q-2)+10), so the remainder is 10. Step 3: If subtraction gives a negative remainder, add the divisor enough times to make it valid.
Correct answer: C. 27
Explanation: Step 1: The condition on the remainder is (0\le r<b). Step 2: If (b=28), the greatest possible value of (r) is 27. Step 3: The remainder can never be equal to the divisor.
Correct answer: A. 23
Explanation: Step 1: Let the number be (25q+24). Step 2: For twice the number, the remainder part is (2\times24=48), and (48=25+23). Step 3: After multiplication, reduce the remainder below 25.
Correct answer: B. 13
Explanation: Step 1: Find the nearest lower multiple of 36 below 589. Step 2: (36\times16=576), so the remainder is (589-576=13). Step 3: For larger numbers, use the nearest lower multiple.
Correct answer: B. 3
Explanation: Step 1: The cube remainder comes from (3^3=27). Step 2: (27=8\times3+3), so the remainder is 3. Step 3: In power questions, keep the calculation small by using the remainder.
Correct answer: B. The remainder can be from 0 to 9
Explanation: Step 1: In Euclid’s lemma, the remainder starts from 0. Step 2: When divided by 10, possible remainders are 0 through 9. Step 3: Do not include the divisor itself in the list of remainders.
Correct answer: B. 6
Explanation: Step 1: The remainder of (p) is 8. Step 2: For (3p-7), the remainder part is (3\times8-7=17), and (17=11+6). Step 3: In a linear expression, always reduce the final remainder below the divisor.
Correct answer: B. 2
Explanation: Step 1: The two remainders are 4 and 5. Step 2: The sum has remainder (4+5=9), and (9=7+2), so the remainder is 2. Step 3: If the sum of remainders is greater than the divisor, reduce it again.
Correct answer: A. 12
Explanation: Step 1: For multiplication, multiply the remainders 10 and 9. Step 2: (10\times9=90), and (90=13\times6+12). Step 3: In products, using remainders instead of whole numbers saves time.
Correct answer: C. 3
Explanation: Step 1: For the cube, consider (2^3=8). Step 2: (8=5\times1+3), so the cube leaves remainder 3. Step 3: In powers, first raise the small remainder to the power.
Correct answer: A. Only 0 and 1
Explanation: Step 1: The possible remainders of a number are 0, 1, 2, and 3. Step 2: The square remainders are respectively 0, 1, 0, and 1. Step 3: A square divided by 4 never leaves remainder 2 or 3.
Correct answer: A. 0
Explanation: Step 1: The remainder of (N) is 11. Step 2: Adding 25 gives total remainder (11+25=36), and 36 is exactly divisible by 18. Step 3: After addition, divide the total remainder again by the divisor.
Correct answer: C. 19
Explanation: Step 1: When the divisor is 19, the remainder can be from 0 to 18. Step 2: 19 is equal to the divisor, so it cannot be a remainder. Step 3: In remainder-range questions, watch carefully for the option equal to the divisor.
Correct answer: A. 10
Explanation: Step 1: (53\times27=1431). Step 2: (1441-1431=10), so the remainder is 10. Step 3: While dividing, choose the multiple that does not exceed the number.
Correct answer: A. 5
Explanation: Step 1: The remainder of (t) is 9. Step 2: The remainder part of (2t+3) is (2\times9+3=21), and (21=16+5). Step 3: Do not forget to reduce the final remainder below 16.
Correct answer: C. 15
Explanation: Step 1: Let the number be (20q+3). Step 2: For five times the number, the remainder is (5\times3=15), which is less than 20. Step 3: In multiplication, multiply the remainder and check the limit.
