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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. Because division by 7 gives remainders from 0 to 6 in a cycle
Explanation: Step 1: The possible remainders on division by 7 are 0, 1, 2, 3, 4, 5, and 6. Step 2: Seven consecutive integers cover all these remainders once. Step 3: The number with remainder 0 is divisible by 7.
Correct answer: B. 1
Explanation: Step 1: The remainders in the two forms are 3 and 5. Step 2: (3^2=9) and (5^2=25), and both leave remainder 1 on division by 8. Step 3: Checking squares of odd remainders modulo 8 is a quick method.
Correct answer: A. 0
Explanation: Step 1: (a-103=59q+44-103=59q-59). Step 2: This can be written as (59(q-1)+0). Step 3: Therefore, the number is exactly divisible by 59 and the remainder is 0.
Correct answer: A. 16
Explanation: Step 1: For the square, take (17^2=289). Step 2: (289=21\times13+16), so the remainder is 16. Step 3: The final answer must be smaller than 21, so 289 cannot be the answer.
Correct answer: A. 0
Explanation: Step 1: The remainder of (x) is 9. Step 2: The remainder of (x^2+x) comes from (9^2+9=90). Step 3: Since 90 is exactly divisible by 10, the remainder is 0.
Correct answer: A. (q=22, r=68)
Explanation: Step 1: (89\times22=1958) and (89\times23=2047). Step 2: The nearest lower multiple below 2026 is 1958, so the remainder is (2026-1958=68). Step 3: Do not accept a negative remainder or a remainder greater than 89.
Correct answer: A. 1899
Explanation: Step 1: Number (=) divisor (\times) quotient (+) remainder. Step 2: (76\times24+75=1824+75=1899). Step 3: The remainder 75 is less than divisor 76, so this form is valid.
Correct answer: A. 18
Explanation: Step 1: For the difference, the remainder is (5-14=-9). Step 2: Add 27 to make it a valid remainder, giving 18. Step 3: In subtraction, add the divisor when the remainder becomes negative.
Correct answer: A. Because division by 8 gives remainders from 0 to 7 in a cycle
Explanation: Step 1: On division by 8, possible remainders are from 0 to 7. Step 2: Eight consecutive integers cover all these remainders once. Step 3: The number with remainder 0 is divisible by 8.
Correct answer: A. 0
Explanation: Step 1: The original remainder is 43. Step 2: 133 leaves remainder 1 when divided by 44. Step 3: The total remainder is (43+1=44), so the final remainder is 0.
Correct answer: B. 11
Explanation: Step 1: The remainder of (u) is 8. Step 2: The remainder of (u^2-u) comes from (8^2-8=56). Step 3: (56=15\times3+11), so the remainder is 11.
Correct answer: D. 9
Explanation: Step 1: (7^2=49), and 49 leaves remainder 10 when divided by 13. Step 2: For (7^4), check (10^2=100). Step 3: (100=13\times7+9), so the remainder is 9.
Correct answer: A. 0
Explanation: Step 1: The original remainder is 17. Step 2: 31 leaves remainder 7 when divided by 24. Step 3: (17+7=24), so the new remainder is 0.
Correct answer: A. 5
Explanation: Step 1: The remainders of (a) and (b) are 11 and 14. Step 2: The remainder of (ab+a) comes from (11\times14+11=165). Step 3: (165=16\times10+5), so the final remainder is 5.
Correct answer: A. (5001=500\times10+1)
Explanation: Step 1: In standard form, the remainder must be from 0 to 499. Step 2: (500\times10=5000), so (5001=5000+1). Step 3: A negative remainder or a remainder greater than 500 does not make the standard Euclidean form.
Correct answer: A. Only 1 and 9
Explanation: Step 1: Check the squares of the given remainders. Step 2: (1^2) and (9^2) leave remainder 1, while (3^2) and (7^2) leave remainder 9. Step 3: When listing possible remainders, count repeated results only once.
Correct answer: B. 10
Explanation: Step 1: For eleven times the number, the remainder part is (11\times8=88). Step 2: (88=39\times2+10), so the final remainder is 10. Step 3: After multiplication, divide the result again by the divisor.
Correct answer: B. 6
Explanation: Step 1: The remainder of (a) is 7. Step 2: The remainder of (a^2+5) comes from (7^2+5=54). Step 3: (54=12\times4+6), so the final remainder is 6.
Correct answer: C. 7
Explanation: Step 1: In (4a+3b), the remainder part is (4\times9+3\times13=75). Step 2: (75=17\times4+7), so the remainder is 7. Step 3: In multi-term expressions, handle each term’s remainder separately.
Correct answer: A. 0
Explanation: Step 1: The original remainder is 57. Step 2: 175 leaves remainder 1 when divided by 58. Step 3: (57+1=58), so the final remainder is 0.
Correct answer: B. 1
Explanation: Step 1: Replace (n) by its remainder 4. Step 2: The remainder of (n^2+n+1) comes from (4^2+4+1=21). Step 3: Dividing 21 by 5 gives remainder 1.
Correct answer: C. 38
Explanation: Step 1: The remainder of (a) is 31. Step 2: For (8a), check (8\times31=248). Step 3: (248=42\times5+38), so the remainder is 38.
Correct answer: C. 97
Explanation: Step 1: The remainder must satisfy (0\le r<97). Step 2: 97 is equal to the divisor, so it cannot be a valid remainder. Step 3: In definition questions, check the remainder range first.
Correct answer: A. 0
Explanation: Step 1: (n^2+2n+1=(n+1)^2). Step 2: Since the remainder of (n) is 13, the remainder of (n+1) is 0. Step 3: When (n+1) is divisible by 14, its square is also divisible by 14.
Correct answer: A. 0
Explanation: Step 1: Add the remainders: (29+31+6=66). Step 2: 66 is exactly divisible by 33. Step 3: Therefore, the final remainder is 0; adding only the remainders is a quick method for multi-term expressions.
