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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: B. (4(q+1))
Explanation: Step 1: Add 1 to (m=4q+3). Step 2: (m+1=4q+4=4(q+1)), so it is divisible by 4. Step 3: When the remainder 3 gets 1 added, it reaches the next multiple of 4.
Correct answer: A. 449
Explanation: Step 1: Number (=) divisor (\times) quotient (+) remainder. Step 2: (18\times24+17=432+17=449). Step 3: Finally check that remainder 17 is less than divisor 18.
Correct answer: B. 651
Explanation: Step 1: Number (=21\times31+r), where (0\le r<21). Step 2: For the least value, take (r=0), so the number is (21\times31=651). Step 3: For the least possible number, use remainder zero.
Correct answer: A. 671
Explanation: Step 1: The number is (21\times31+r). Step 2: The greatest value of (r) is 20, so the number is (651+20=671). Step 3: In such questions, take the maximum remainder as one less than the divisor.
Correct answer: B. (q) integer and (0\le r<b)
Explanation: Step 1: In the lemma, (q) is an integer and (r) is the remainder. Step 2: The key condition is (0\le r<b). Step 3: In definition-based questions, the remainder condition is the most important clue.
Correct answer: A. Only 0 and 1
Explanation: Step 1: A number can be (3q), (3q+1), or (3q+2). Step 2: The square remainders are 0, 1, and the remainder of (2^2=4), which is 1. Step 3: For squares modulo 3, remainder 2 never appears.
Correct answer: D. 6
Explanation: Step 1: On division by 5, the remainder can be 0, 1, 2, 3, or 4. Step 2: 6 is greater than divisor 5, so it cannot be a remainder. Step 3: First eliminate options equal to or greater than the divisor.
Correct answer: A. 0
Explanation: Step 1: The remainder of (N) is 11. Step 2: The remainder of 25 on division by 12 is 1, so total remainder (11+1=12), which becomes 0. Step 3: In addition, add the remainders and then reduce by the divisor.
Correct answer: B. 5
Explanation: Step 1: Let (y=9q+2). Step 2: (5y+4=45q+10+4=45q+14=9(5q+1)+5). Step 3: For a linear expression, replace the number by its remainder and simplify.
Correct answer: C. 11
Explanation: Step 1: The number is (14q+13). Step 2: Subtracting 2 gives (14q+11), so the remainder is 11. Step 3: In subtraction, if the remainder does not become negative, subtract directly.
Correct answer: B. 1
Explanation: Step 1: Let (z=8q+1). Step 2: Use the remainder in the power: (1^3=1), so the remainder on division by 8 is still 1. Step 3: For powers, you do not need to expand the whole expression.
Correct answer: A. 1
Explanation: Step 1: Let the number be (16q+15). Step 2: The square remainder is obtained from (15^2=225) divided by 16. (225=16\times14+1). Step 3: The square of a remainder (b-1) often gives remainder 1.
Correct answer: C. (7q,7q+1,\ldots,7q+6)
Explanation: Step 1: On division by 7, remainders can be from 0 to 6. Step 2: So the forms are from (7q) to (7q+6). Step 3: Include remainder 0 and do not include remainder 7 in the complete list.
Correct answer: A. 0
Explanation: Step 1: The remainder of (a) is 19. Step 2: Adding 1 gives (20q+20=20(q+1)), so the remainder is 0. Step 3: Adding 1 to a remainder one less than the divisor gives the next multiple.
Correct answer: B. 2
Explanation: Step 1: For the cube, work with the remainder (2^3=8). Step 2: (8=6\times1+2), so the cube leaves remainder 2. Step 3: In powers, keep the calculation small by using the remainder.
Correct answer: B. 12
Explanation: Step 1: The remainders of the two numbers are 5 and 7. Step 2: Their sum gives remainder (5+7=12), which is less than 18. Step 3: For a sum, adding the remainders first is easier.
Correct answer: B. 7
Explanation: Step 1: Add the remainders 14 and 11. Step 2: (14+11=25), and (25=18+7), so the remainder is 7. Step 3: If the sum of remainders exceeds the divisor, reduce it again.
Correct answer: A. 2
Explanation: Step 1: In multiplication, multiply the remainders. Step 2: (3\times4=12), and 12 leaves remainder 2 on division by 10. Step 3: For products, multiply the remainders instead of the whole numbers.
Correct answer: A. 10
Explanation: Step 1: The number is (13q+12). Step 2: For three times the number, the remainder part is (3\times12=36), and (36=13\times2+10). Step 3: After multiplication, reduce the remainder below the divisor.
Correct answer: B. 1
Explanation: Step 1: The possible remainders are 1 and 3. Step 2: (1^2=1) and (3^2=9=4\times2+1), so the remainder is 1 in both cases. Step 3: The square of an odd number leaves remainder 1 when divided by 4.
Correct answer: B. 1
Explanation: Step 1: The square remainder comes from dividing (4^2=16) by 5. Step 2: (16=5\times3+1), so the remainder is 1. Step 3: A remainder one less than the divisor often gives square remainder 1.
Correct answer: A. 3
Explanation: Step 1: Write (a=7q+5). Step 2: (a-9=7q-4=7(q-1)+3), so the remainder is 3. Step 3: If subtraction gives a negative remainder, add the divisor to make it valid.
Correct answer: B. 3
Explanation: Step 1: The number is (9q+8). Step 2: Adding 4 gives (9q+12=9(q+1)+3), so the remainder is 3. Step 3: Reduce the new remainder below the divisor by subtracting 9.
Correct answer: A. 8
Explanation: Step 1: Find the nearest lower multiple of 45 below 728. Step 2: (45\times16=720), so (728-720=8). Step 3: The nearest-multiple method saves time in large divisions.
Correct answer: A. 0
Explanation: Step 1: The number is (12q+7). Step 2: Adding 29 gives total remainder (7+29=36), and 36 is exactly divisible by 12. Step 3: You may add the given number directly, then find the final remainder.
