Muft Shiksha™ एक 100% Free Education Portal है 🇮🇳, जिसका उद्देश्य Class 9–12 के हर विद्यार्थी तक High-Quality Education को पूरी तरह मुफ्त पहुँचाना है। 🇮🇳 हम मानते हैं कि अच्छी शिक्षा किसी student की आर्थिक स्थिति पर निर्भर नहीं होनी चाहिए। 🇮🇳 हर विद्यार्थी को वही Quality Study Material, MCQs, Quizzes, Exam Preparation, Concept-Based Learning और Bilingual Support मिलना चाहिए, जो आमतौर पर महंगी Coaching या Premium Platforms में मिलता है। Muft Shiksha™ 🇮🇳 इसी सोच के साथ बनाया गया है
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. Because division by 5 gives the cycle of remainders 0, 1, 2, 3, 4
Explanation: Step 1: Any integer divided by 5 has one of the forms (5q), (5q+1), (5q+2), (5q+3), or (5q+4). Step 2: Five consecutive integers cover all five remainders. Step 3: The number with remainder 0 is divisible by 5.
Correct answer: B. 1
Explanation: Step 1: The possible remainders are 1 and 5. Step 2: (1^2=1) and (5^2=25=6\times4+1), so the remainder is 1 in both cases. Step 3: In such forms, work only with the remainder, not the whole number.
Correct answer: C. 18
Explanation: Step 1: (a-48=37q+29-48=37q-19). Step 2: This can be written as (37(q-1)+18), so the remainder is 18. Step 3: Add the divisor once to make a negative remainder valid.
Correct answer: A. 1
Explanation: Step 1: For the square, take (11^2=121). Step 2: (121=15\times8+1), so the remainder is 1. Step 3: Do not write 121 as the final answer because a remainder must be smaller than the divisor.
Correct answer: A. 0
Explanation: Step 1: The remainder of (x) is 7. Step 2: The remainder of (x^2+x) comes from (7^2+7=56), and 56 is divisible by 8. Step 3: Substituting the remainder in the expression gives a quick solution.
Correct answer: A. (q=22, r=5)
Explanation: Step 1: (23\times22=506) and (23\times23=529). Step 2: The nearest lower multiple of 23 below 511 is 506, so the remainder is 5. Step 3: Do not accept a negative remainder or a remainder greater than the divisor.
Correct answer: A. 764
Explanation: Step 1: Number (=) divisor (\times) quotient (+) remainder. Step 2: (45\times16+44=720+44=764). Step 3: The remainder 44 is less than divisor 45, so the form is valid.
Correct answer: A. 10
Explanation: Step 1: For the difference, the remainder is (3-9=-6). Step 2: Add 16 to make it valid, giving 10. Step 3: In subtraction, add the divisor when the remainder becomes negative.
Correct answer: A. Because division by 6 gives the cycle of remainders 0, 1, 2, 3, 4, 5
Explanation: Step 1: On division by 6, possible remainders are from 0 to 5. Step 2: Six consecutive integers cover all these remainders once. Step 3: The number with remainder 0 is divisible by 6.
Correct answer: A. 0
Explanation: Step 1: The original remainder is 34. Step 2: (71) leaves remainder 1 on division by 35, so total remainder (34+1=35), which becomes 0. Step 3: First reduce the large added number to a small remainder.
Correct answer: A. 8
Explanation: Step 1: The remainder of (u) is 5. Step 2: The remainder of (u^2-u) comes from (5^2-5=20), and (20=12+8). Step 3: Substitute the remainder first, then find the final remainder.
Correct answer: B. 4
Explanation: Step 1: (7^2=49), which leaves remainder 4 on division by 9. Step 2: For (7^4), use (4^2=16), and 16 leaves remainder 7 on division by 9. Step 3: In higher powers, reduce the remainder after each step.
Correct answer: A. 0
Explanation: Step 1: The number is (18q+12). Step 2: Adding 6 gives (18q+18=18(q+1)), so the remainder is 0. Step 3: When the remainder and the added number make the divisor, the new remainder becomes zero.
Correct answer: A. 7
Explanation: Step 1: The remainders of (a) and (b) are 9 and 6. Step 2: The remainder of (ab+a) comes from (9\times6+9=63), and (63=14\times4+7). Step 3: In a mixed expression, handle the remainder of each term separately.
Correct answer: A. (1201=120\times10+1)
Explanation: Step 1: In standard form, the remainder must be from 0 to 119. Step 2: (120\times10=1200), so (1201=1200+1). Step 3: Along with correct calculation, the valid range of the remainder is necessary.
Correct answer: B. 1
Explanation: Step 1: These remainders all represent odd numbers. Step 2: (1^2,3^2,5^2,7^2) all leave remainder 1 when divided by 8. Step 3: Remember that the square of an odd number leaves remainder 1 on division by 8.
Correct answer: B. 13
Explanation: Step 1: For eight times the number, the remainder part is (8\times5=40). Step 2: (40=27+13), so the final remainder is 13. Step 3: After multiplication, reduce the result below the divisor.
Correct answer: A. 4
Explanation: Step 1: The remainder of (a) is 3. Step 2: The remainder of (a^2+2) comes from (3^2+2=11), and (11=7+4). Step 3: Substitute the remainder in the expression, then find the final remainder.
Correct answer: A. 4
Explanation: Step 1: In (3a+b), the remainder part is (3\times6+8=26). Step 2: (26=11\times2+4), so the remainder is 4. Step 3: In multi-term expressions, handle the remainder of each term separately.
Correct answer: A. 0
Explanation: Step 1: The original remainder is 41. Step 2: (85) leaves remainder 1 on division by 42, so total remainder (41+1=42), which becomes 0. Step 3: When adding a large number, first find its smaller remainder.
Correct answer: A. 1
Explanation: Step 1: Replace (n) by its remainder 3. Step 2: The remainder of (n^2+n+1) comes from (3^2+3+1=13), and 13 leaves remainder 1 when divided by 4. Step 3: In polynomial-like expressions, substituting the remainder makes the solution direct.
Correct answer: A. 18
Explanation: Step 1: The remainder of (a) is 23. Step 2: For (6a), compute (6\times23=138), and (138=30\times4+18). Step 3: After multiplication, reduce the answer below the divisor.
Correct answer: C. 63
Explanation: Step 1: The remainder must satisfy (0\le r<63). Step 2: 63 is equal to the divisor, so it cannot be a valid remainder. Step 3: In definition-based questions, check the condition (r<b) first.
Correct answer: A. 0
Explanation: Step 1: (n^2+2n+1=(n+1)^2). Step 2: Since (n) has remainder 9, (n+1) has remainder 0, so its square is divisible by 10. Step 3: First recognize the structure of the expression to reduce calculation.
Correct answer: B. 1
Explanation: Step 1: Add the remainders: (17+19+6=42). Step 2: (42) is exactly divisible by 21, so the remainder should be 0. Step 3: In multi-term questions, add only the remainders and reduce at the end.
