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 one of them is divisible by 2 and one is divisible by 3
Explanation: Step 1: Among two consecutive integers, one is even, so a factor 2 is present. Step 2: Among three consecutive integers, one is divisible by 3, so a factor 3 is present. Step 3: Since 2 and 3 together make 6, the product is divisible by 6.
Correct answer: B. 1
Explanation: Step 1: A number of the form (4q+1) is odd. Step 2: The square of an odd number leaves remainder 1 when divided by 8; for example, (1^2) and (5^2) both leave remainder 1. Step 3: Remember the rule that odd squares leave remainder 1 on division by 8.
Correct answer: A. 14
Explanation: Step 1: (a-35=29q+20-35=29q-15). Step 2: This can be written as (29(q-1)+14), so the remainder is 14. Step 3: Add the divisor to a negative remainder to make it valid.
Correct answer: C. 12
Explanation: Step 1: For the square, take (8^2=64). Step 2: (64=13\times4+12), so the remainder is 12. Step 3: Do not write 64 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 5. Step 2: The remainder of (x^2+x) comes from (5^2+5=30), and 30 is divisible by 6. Step 3: Substitute the remainder in the expression to solve quickly.
Correct answer: A. (q=20, r=12)
Explanation: Step 1: (17\times20=340) and (17\times21=357). Step 2: The nearest lower multiple of 17 below 352 is 340, so the remainder is 12. Step 3: Do not accept a negative remainder or a remainder greater than 17.
Correct answer: A. 511
Explanation: Step 1: Number (=) divisor (\times) quotient (+) remainder. Step 2: (32\times15+31=480+31=511). Step 3: Remainder 31 is less than divisor 32, so the form is valid.
Correct answer: A. 13
Explanation: Step 1: For the difference, the remainder is (9-10=-1). Step 2: Add 14 to make it a valid remainder, giving 13. Step 3: In subtraction, add the divisor when the remainder becomes negative.
Correct answer: A. Because division by 4 gives the cycle of remainders 0, 1, 2, 3
Explanation: Step 1: Any integer divided by 4 is of the form (4q), (4q+1), (4q+2), or (4q+3). Step 2: Four consecutive integers cover all these four remainders. Step 3: The one with remainder 0 is divisible by 4.
Correct answer: A. 0
Explanation: Step 1: The original remainder is 26. Step 2: Adding 28 gives total remainder (26+28=54), which is exactly divisible by 27. Step 3: It is easier to reduce the added number by the divisor and combine remainders.
Correct answer: B. 2
Explanation: Step 1: The remainder of (u) is 7. Step 2: The remainder of (u^2-u) comes from (7^2-7=42), and (42=10\times4+2). Step 3: Substitute the remainder first, then find the final remainder.
Correct answer: D. 4
Explanation: Step 1: For the fourth power, look at the remainder of (4^4). Step 2: (4^2=16), which leaves remainder 2 when divided by 7; then (4^4) leaves the same remainder as (2^2=4). Step 3: For higher powers, reduce remainders step by step.
Correct answer: A. 0
Explanation: Step 1: The number is (15q+10). Step 2: Adding 5 gives (15q+15=15(q+1)), so the remainder is 0. Step 3: When the remainder and added number together make the divisor, the new remainder becomes 0.
Correct answer: A. 0
Explanation: Step 1: The remainders of (a) and (b) are 8 and 5. Step 2: The remainder of (ab+a) comes from (8\times5+8=48), and 48 is divisible by 12. Step 3: In a mixed expression, find the remainder of each part separately.
Correct answer: B. (999=100\times9+99)
Explanation: Step 1: In standard form, the remainder is not negative. Step 2: (100\times9=900), so (999=900+99), and 99 is less than 100. Step 3: A form with a negative remainder may look computationally close, but it is not Euclidean form.
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: Squares of many numbers coprime to 6 show remainder 1, but checking is always better.
Correct answer: B. 6
Explanation: Step 1: For seven times the number, the remainder part is (7\times4=28). Step 2: (28=22+6), so the final remainder is 6. Step 3: If the result after multiplication is greater than the divisor, reduce it.
Correct answer: A. 0
Explanation: Step 1: The remainder of (a) is 2. Step 2: The remainder of (a^2+1) comes from (2^2+1=5), which is exactly divisible by 5. Step 3: After substituting the remainder in the expression, check it again by the divisor.
Correct answer: B. 6
Explanation: Step 1: In (2a+b), the remainder part is (2\times4+7=15). Step 2: (15=9+6), so the remainder is 6. Step 3: In such questions, handle the remainder of each term separately.
Correct answer: A. 0
Explanation: Step 1: The original remainder is 29. Step 2: 61 leaves remainder 1 when divided by 30, so total remainder (29+1=30), which becomes 0. Step 3: It is useful to reduce a large added number to a smaller remainder first.
Correct answer: B. 1
Explanation: Step 1: Replace (n) by its remainder 2. Step 2: The remainder of (n^2+n+1) comes from (2^2+2+1=7), and 7 leaves remainder 1 when divided by 3. Step 3: In polynomial-like expressions, substituting the remainder makes the solution simple.
Correct answer: A. 13
Explanation: Step 1: The remainder of (a) is 17. Step 2: For (5a), compute (5\times17=85), and (85=24\times3+13). Step 3: After multiplication, the remainder must be reduced below the divisor.
Correct answer: C. 41
Explanation: Step 1: The remainder must satisfy (0\le r<41). Step 2: 41 is equal to the divisor, so it cannot be a remainder. Step 3: In definition-based questions, (r<b) is the most important rule.
Correct answer: A. 0
Explanation: Step 1: The expression (n^2+2n+1) is ((n+1)^2). Step 2: Since (n) has remainder 7, (n+1) has remainder 0, so its square is divisible by 8. Step 3: Recognizing the structure of the expression reduces calculation.
Correct answer: A. 0
Explanation: Step 1: Add the remainders: (13+16+7=36). Step 2: 36 is exactly divisible by 18, so the remainder is 0. Step 3: In multi-term questions, add only the remainders and then reduce at the end.
