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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.
TOPIC PRACTICE
Up to 25 questions from this page. Select your focus, then start.
Step 1: (12 \times 13=156) and (12 \times 14=168). Step 2: (156) is the nearest smaller multiple of (12), so the remainder is (1). Step 3: In the correct answer, the remainder must be less than the divisor and not negative.
Step 1: In Euclid’s division lemma, the remainder may start from (0). Step 2: The remainder is always less than the divisor (21), so (0 \le r < 21) is correct. Step 3: Never take the remainder equal to the divisor.
Step 1: To find the dividend, use (a=bq+r). Step 2: (a=17 \times 11+9=187+9=196). Step 3: Matching words with symbols before calculation reduces mistakes.
Step 1: On division by (13), remainders can be from (0) to (12). Step 2: The total number of these values is (13). Step 3: For a divisor (b), the number of possible remainders is (b).
Step 1: (19 \times 12=228) and (19 \times 13=247). Step 2: Since (247) is greater, the remainder is (246-228=18). Step 3: If the next multiple is greater, use the previous multiple.
Step 1: (24 \times 12=288) and (24 \times 13=312). Step 2: Since (312) is greater, (305=24 \times 12+17) is correct. Step 3: A form with a negative remainder is not the standard Euclidean form.
Step 1: (25) cannot be the remainder because it is greater than (18). Step 2: (25=18+7), so (18q+25=18(q+1)+7). Step 3: Divide a large remainder again by the divisor to bring it into the correct range.
Step 1: (36) is twice (18). Step 2: (18q+36=18(q+2)+0), so the remainder is (0). Step 3: If the added part is a multiple of the divisor, the remainder can become zero.
Step 1: In (a=10q+9), the remainder is (9). Step 2: (a+4=10q+13=10(q+1)+3), so the new remainder is (3). Step 3: If the sum of remainders exceeds the divisor, subtract the divisor.
Step 1: Add (20) to the old remainder (3) to get (23). Step 2: (23=11 \times 2+1), so the new remainder is (1). Step 3: In addition-based questions, divide the new sum by the same divisor again.
Step 1: (35 \times 12=420). Step 2: (437-420=17), so the remainder is (17). Step 3: After getting the answer, check that (17<35).
Step 1: (35 \times 12=420) and (35 \times 13=455). Step 2: Since (455) is greater than (437), the quotient is (12). Step 3: Checking the next multiple helps in choosing the quotient.
Step 1: On division by (8), remainders can be from (0) to (7). Step 2: So in (8q+r), (r=0,1,2,3,4,5,6,7). Step 3: Do not include (8q+8) while writing standard forms.
Step 1: (9 \times 7=63) and (9 \times 8=72). Step 2: (63) is the correct nearest smaller multiple, so the remainder is (64-63=1). Step 3: The remainder must be less than (9).
Step 1: In (a=7q+6), the remainder is (6). Step 2: (a+3=7q+9=7(q+1)+2), so the remainder is (2). Step 3: If the new remainder exceeds the divisor, subtract the divisor to get the answer.
Step 1: Adding (29) to the old remainder (2) gives (31). Step 2: (31=15 \times 2+1), so the new remainder is (1). Step 3: In large addition cases, it is enough to find the remainder of the new sum.
Step 1: In (a=20q+17), the remainder is (17). Step 2: (a-5=20q+12), so the remainder is (12). Step 3: In subtraction-based questions, subtract the given number from the old remainder.
Step 1: (a-5=20q-2), but the remainder cannot be negative. Step 2: (20q-2=20(q-1)+18), so the correct remainder is (18). Step 3: If a negative remainder appears, reduce the quotient by one and make the remainder positive.
Step 1: (23 \times 23=529). Step 2: The number divides exactly, so the remainder is (0). Step 3: Recognizing exact division makes the calculation faster.
Step 1: (23 \times 23=529). Step 2: The product equals the dividend, so the quotient is (23). Step 3: When the remainder is (0), the chosen multiplier is the correct quotient.
Step 1: (22) is a multiple of (11). Step 2: (11q+22=11(q+2)+0), so the correct remainder is (0). Step 3: If the added part is exactly divisible by the divisor, the remainder becomes zero.
Step 1: Divide (27) by (11). Step 2: (27=11 \times 2+5), so (11q+27=11(q+2)+5). Step 3: Finding the remainder of the large added part separately is an easy method.
Step 1: The number can be written as (16q+15). Step 2: The next number is (16q+16=16(q+1)+0). Step 3: After the greatest remainder, the next number has remainder (0).
Step 1: The number has the form (16q). Step 2: The next number is (16q+1), so the remainder is (1). Step 3: In consecutive numbers, remainders move in order like (0,1,2).
Step 1: In the Euclidean form (a=bq+r), (b) is the divisor. Step 2: In (618=47 \times 13+7), (47) is the number used for division. Step 3: Carefully identify the first number written in the product.
QUIZ COMPLETE
