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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: Find the nearest lower multiple of 238 below 4961. Step 2: (238\times20=4760), so the remainder is (4961-4760=201). Step 3: Since the remainder is smaller than 238, this is the valid Euclidean form.
Step 1: The number is of the form (87\times46+r), where (0\le r<87). Step 2: The greatest remainder is 86, so the number is (4002+86=4088). Step 3: For the greatest value, the remainder is always one less than the divisor.
Step 1: The number is (143\times23+r). Step 2: For the least value, the remainder is 0, so the number is (143\times23=3289). Step 3: In least-value questions, taking the remainder as zero is the clearest method.
Step 1: In standard form, the remainder must be from 0 to 390. Step 2: (391\times15=5865), so (6127=5865+262). Step 3: A negative remainder or a remainder greater than 391 does not make the correct Euclidean form.
Step 1: The remainder of (x) is 64. Step 2: 140 leaves remainder 6 when divided by 67, so the total remainder is (64+6=70). Step 3: Since (70=67+3), the final remainder is 3.
Step 1: Replace (n) by its remainder 31. Step 2: The remainder of (9n+22) comes from (9\times31+22=301). Step 3: Since (301=43\times7+0), the remainder is 0.
Step 1: The square remainder comes from dividing (16^2=256) by 19. Step 2: (256=19\times13+9), so the remainder is 9. Step 3: In square questions, square only the remainder instead of the full number.
Step 1: On division by 17, remainders can be from 0 to 16. Step 2: Therefore, all forms are from (17q) to (17q+16). Step 3: Include zero remainder, but do not include 17.
Step 1: The remainder of (a) is 96, one less than 97. Step 2: Adding 1 gives (97q+97=97(q+1)). Step 3: The number becomes exactly divisible by 97, so the remainder is 0.
Step 1: Write (m=53q+12). Step 2: (m-118=53q+12-118=53q-106=53(q-2)). Step 3: It is exactly divisible by 53, so the remainder is 0.
Step 1: The remainder condition is (0\le r<144). Step 2: The greatest integer smaller than 144 is 143. Step 3: The remainder is never equal to the divisor.
Step 1: For nine times the number, the remainder part is (9\times58=522). Step 2: (522=59\times8+50), so the remainder is 50. Step 3: After multiplication, always reduce the result below the divisor.
Step 1: Find the nearest lower multiple of 359 below 9264. Step 2: (359\times25=8975), so the remainder is (9264-8975=289). Step 3: With large numbers, the nearest lower multiple method saves time.
Step 1: For the cube, check (11^3=1331). Step 2: (1331=18\times73+17), so the remainder is 17. Step 3: In power questions, reduce remainders repeatedly to keep the calculation simple.
Step 1: In Euclid’s lemma, (0\le r<b). Step 2: Here (b=32), so the remainder can be from 0 to 31. Step 3: Include 0 in the list of remainders and do not include 32.
Step 1: The remainder of (p) is 19. Step 2: The remainder of (7p-18) comes from (7\times19-18=115). Step 3: Since (115=23\times5), the final remainder is 0.
Step 1: The two remainders are 25 and 28. Step 2: The sum remainder comes from (25+28=53). Step 3: Since (53=31+22), the final remainder is 22.
Step 1: For multiplication, multiply the remainders 32 and 35. Step 2: (32\times35=1120), and (1120=37\times30+10). Step 3: In product questions, the answer is the final remainder, not the full product.
Step 1: For the cube, check (6^3=216). Step 2: (216=13\times16+8), so the remainder is 8. Step 3: In powers, working with small remainders avoids large calculations.
Step 1: On division by 11, remainders can be from 0 to 10. Step 2: Their distinct square remainders are 0, 1, 3, 4, 5, and 9. Step 3: In square-remainder questions, making a short list of possible remainders is useful.
Step 1: The remainder of (N) is 39. Step 2: 102 leaves remainder 8 when divided by 47. Step 3: The total remainder is (39+8=47), so the final remainder is 0.
Step 1: When the divisor is 69, the remainder can be from 0 to 68. Step 2: 69 is equal to the divisor, so it cannot be a remainder. Step 3: In definition-based questions, check the condition (r<b) first.
Step 1: (412\times23=9476) and (412\times24=9888). Step 2: The nearest lower multiple below 9876 is 9476, so the remainder is (400). Step 3: Since the remainder is less than 412, it is valid.
Step 1: The remainder of (t) is 37. Step 2: The remainder part of (6t+17) is (6\times37+17=239). Step 3: (239=46\times5+9), so the final remainder is 9.
Step 1: For thirteen times the number, the remainder part is (13\times15=195). Step 2: (195=73\times2+49), so the final remainder is 49. Step 3: After multiplication, reduce the result by the divisor to make a valid remainder.
QUIZ COMPLETE
