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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 9 gives remainders from 0 to 8 in a cycle
Explanation: Step 1: On division by 9, possible remainders are from 0 to 8. Step 2: Nine consecutive integers cover all these remainders once. Step 3: The number with remainder 0 is divisible by 9.
Correct answer: D. 9
Explanation: Step 1: The remainders in the two forms are 3 and 7. Step 2: (3^2=9) and (7^2=49), and both leave remainder 9 when divided by 10. Step 3: In form-based questions, work only with the remainder.
Correct answer: A. 0
Explanation: Step 1: (a-119=67q+52-119=67q-67). Step 2: This can be written as (67(q-1)+0). Step 3: The number is exactly divisible by 67, so the remainder is 0.
Correct answer: B. 23
Explanation: Step 1: For the square, take (21^2=441). Step 2: (441=26\times16+25), so the remainder should be 25. Step 3: The final answer must always be less than 26.
Correct answer: A. 0
Explanation: Step 1: The remainder of (x) is 11. Step 2: The remainder of (x^2+x) comes from (11^2+11=132). Step 3: Since 132 is exactly divisible by 12, the remainder is 0.
Correct answer: A. (q=29, r=124)
Explanation: Step 1: (137\times29=3973) and (137\times30=4110). Step 2: The nearest lower multiple below 4097 is 3973, so the remainder is (4097-3973=124). Step 3: Do not accept a negative remainder or a remainder greater than 137.
Correct answer: A. 3071
Explanation: Step 1: Number (=) divisor (\times) quotient (+) remainder. Step 2: (96\times31+95=2976+95=3071). Step 3: The remainder 95 is less than divisor 96, so the form is valid.
Correct answer: A. 24
Explanation: Step 1: For the difference, the remainder is (8-19=-11). Step 2: Add 35 to make it a valid remainder, giving 24. Step 3: In subtraction, add the divisor when the remainder becomes negative.
Correct answer: A. Because division by 10 gives remainders from 0 to 9 in a cycle
Explanation: Step 1: On division by 10, possible remainders are from 0 to 9. Step 2: Ten consecutive integers cover all these remainders once. Step 3: The number with remainder 0 is divisible by 10.
Correct answer: A. 0
Explanation: Step 1: The original remainder is 61. Step 2: 187 leaves remainder 1 when divided by 62. Step 3: The total remainder is (61+1=62), so the final remainder is 0.
Correct answer: B. 2
Explanation: Step 1: The remainder of (u) is 11. Step 2: The remainder of (u^2-u) comes from (11^2-11=110). Step 3: (110=18\times6+2), so the remainder is 2.
Correct answer: C. 4
Explanation: Step 1: (10^2=100), and 100 leaves remainder 15 when divided by 17. Step 2: For (10^4), check (15^2=225). Step 3: (225=17\times13+4), so the remainder is 4.
Correct answer: A. 0
Explanation: Step 1: The original remainder is 29. Step 2: 43 leaves remainder 7 when divided by 36. Step 3: (29+7=36), so the new remainder is 0.
Correct answer: B. 7
Explanation: Step 1: The remainders of (a) and (b) are 13 and 18. Step 2: The remainder of (ab+a) comes from (13\times18+13=247). Step 3: (247=20\times12+7), so the final remainder is 7.
Correct answer: A. (7001=700\times10+1)
Explanation: Step 1: In standard form, the remainder must be from 0 to 699. Step 2: (700\times10=7000), so (7001=7000+1). Step 3: A negative remainder or a remainder greater than 700 does not make the standard Euclidean form.
Correct answer: B. 1
Explanation: Step 1: Check the squares of the given remainders. Step 2: (1^2), (5^2), (7^2), and (11^2) all leave remainder 1 when divided by 12. Step 3: Checking the squares of possible remainders separately is a safe method.
Correct answer: C. 8
Explanation: Step 1: For thirteen times the number, the remainder part is (13\times11=143). Step 2: (143=45\times3+8), so the final remainder is 8. Step 3: After multiplication, divide the result again by the divisor.
Correct answer: C. 8
Explanation: Step 1: The remainder of (a) is 9. Step 2: The remainder of (a^2+7) comes from (9^2+7=88). Step 3: (88=16\times5+8), so the final remainder is 8.
Correct answer: B. 5
Explanation: Step 1: In (5a+4b), the remainder part is (5\times11+4\times16=119). Step 2: (119=19\times6+5), so the remainder is 5. Step 3: In multi-term expressions, handle each term’s remainder separately.
Correct answer: A. 0
Explanation: Step 1: The original remainder is 73. Step 2: 223 leaves remainder 1 when divided by 74. Step 3: (73+1=74), so the final remainder is 0.
Correct answer: A. 1
Explanation: Step 1: Replace (n) by its remainder 5. Step 2: The remainder of (n^2+n+1) comes from (5^2+5+1=31). Step 3: Dividing 31 by 6 gives remainder 1.
Correct answer: B. 34
Explanation: Step 1: The remainder of (a) is 37. Step 2: For (10a), check (10\times37=370). Step 3: (370=48\times7+34), so the remainder is 34.
Correct answer: C. 109
Explanation: Step 1: The remainder must satisfy (0\le r<109). Step 2: 109 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 21, the remainder of (n+1) is 0. Step 3: When (n+1) is divisible by 22, its square is also divisible by 22.
Correct answer: A. 0
Explanation: Step 1: Add the remainders: (35+37+6=78). Step 2: 78 is exactly divisible by 39. Step 3: Therefore, the final remainder is 0; adding only the remainders is a quick method for multi-term expressions.
