H.C.F. and L.C.M. of Numbers: Formulas, Shortcuts and Solved Examples — MCQs with Answers (Set 5)

Let the fractions be 19/23 and 29/12. For fractions, H.C.F. = H.C.F. of numerators over L.C.M. of denominators. H.C.F. of numerators: H.C.F.(19, 29) = 1. L.C.M. of denominators: L.C.M.(23, 12) = 276. So the H.C.F. is 1/276. Reducing by their common factor 1 gives 1/276. Check: 1 x 1 = 1 and 276 x 1 = 276, confirming the reduction.

Let the numbers be 5^4 x 11^3 and 5^3 x 11^4 x 2^1. For each common prime, the H.C.F. uses the lower exponent. For 5: min(4, 3) = 3. For 11: min(3, 4) = 3. For 2: min(0, 1) = 0. Combining these primes gives H.C.F. = 5^3 x 11^3 = 166375. Check: 831875/166375 = 5 and 3660250/166375 = 22, both whole numbers.

Let the numbers be 11^2 x 7^3 and 11^2 x 7^1 x 2^1. For each common prime, the H.C.F. uses the lower exponent. For 11: min(2, 2) = 2. For 7: min(3, 1) = 1. For 2: min(0, 1) = 0. Combining these primes gives H.C.F. = 11^2 x 7^1 = 847. Check: 41503/847 = 49 and 1694/847 = 2, both whole numbers.

Let the numbers be 2^3 x 11^2 x 7^1 and 2^1 x 7^2. For each common prime, the H.C.F. uses the lower exponent. For 2: min(3, 1) = 1. For 11: min(2, 0) = 0. For 7: min(1, 2) = 1. Combining these primes gives H.C.F. = 2^1 x 7^1 = 14. Check: 6776/14 = 484 and 98/14 = 7, both whole numbers.

Let the numbers be 11^2 x 7^1 x 5^2 and 11^4 x 7^4 x 5^3. For each common prime, the H.C.F. uses the lower exponent. For 11: min(2, 4) = 2. For 7: min(1, 4) = 1. For 5: min(2, 3) = 2. Combining these primes gives H.C.F. = 11^2 x 7^1 x 5^2 = 21175. Check: 21175/21175 = 1 and 4394130125/21175 = 207515, both whole numbers.

Let the numbers be 2^3 x 7^3 and 2^4 x 11^2 x 7^2. For each common prime, the H.C.F. uses the lower exponent. For 2: min(3, 4) = 3. For 11: min(0, 2) = 0. For 7: min(3, 2) = 2. Combining these primes gives H.C.F. = 2^3 x 7^2 = 392. Check: 2744/392 = 7 and 94864/392 = 242, both whole numbers.

Let the numbers be 11^3 x 2^4 x 5^3 and 11^4 x 2^3. For each common prime, the H.C.F. uses the lower exponent. For 11: min(3, 4) = 3. For 2: min(4, 3) = 3. For 5: min(3, 0) = 0. Combining these primes gives H.C.F. = 11^3 x 2^3 = 10648. Check: 2662000/10648 = 250 and 117128/10648 = 11, both whole numbers.

Let the numbers be 2^4 x 5^4 and 2^3 x 5^1 x 3^2. For each common prime, the H.C.F. uses the lower exponent. For 2: min(4, 3) = 3. For 5: min(4, 1) = 1. For 3: min(0, 2) = 0. Combining these primes gives H.C.F. = 2^3 x 5^1 = 40. Check: 10000/40 = 250 and 360/40 = 9, both whole numbers.

Let the numbers be 7^4 x 5^1 and 7^1 x 5^2 x 11^3. For each common prime, the H.C.F. uses the lower exponent. For 7: min(4, 1) = 1. For 5: min(1, 2) = 1. For 11: min(0, 3) = 0. Combining these primes gives H.C.F. = 7^1 x 5^1 = 35. Check: 12005/35 = 343 and 232925/35 = 6655, both whole numbers.

Let the numbers be 38, 37, 35. Break each into prime factors: 38 = 2 x 19; 37 = 37; 35 = 5 x 7. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 49210. Check: 49210/38 = 1295, 49210/37 = 1330, 49210/35 = 1406, so 49210 is exactly divisible by each number.

Let the numbers be 31, 44. Break each into prime factors: 31 = 31; 44 = 2 x 2 x 11. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 1364. Check: 1364/31 = 44, 1364/44 = 31, so 1364 is exactly divisible by each number.

Let the numbers be 56, 53, 44. Break each into prime factors: 56 = 2 x 2 x 2 x 7; 53 = 53; 44 = 2 x 2 x 11. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 32648. Check: 32648/56 = 583, 32648/53 = 616, 32648/44 = 742, so 32648 is exactly divisible by each number.

Let the numbers be 51, 24. Break each into prime factors: 51 = 3 x 17; 24 = 2 x 2 x 2 x 3. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 408. Check: 408/51 = 8, 408/24 = 17, so 408 is exactly divisible by each number.

Let the numbers be 24, 28. Break each into prime factors: 24 = 2 x 2 x 2 x 3; 28 = 2 x 2 x 7. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 168. Check: 168/24 = 7, 168/28 = 6, so 168 is exactly divisible by each number.

Let the numbers be 21, 33, 54. Break each into prime factors: 21 = 3 x 7; 33 = 3 x 11; 54 = 2 x 3 x 3 x 3. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 4158. Check: 4158/21 = 198, 4158/33 = 126, 4158/54 = 77, so 4158 is exactly divisible by each number.

Let the numbers be 41, 38. Break each into prime factors: 41 = 41; 38 = 2 x 19. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 1558. Check: 1558/41 = 38, 1558/38 = 41, so 1558 is exactly divisible by each number.

Let the numbers be 53, 46. Break each into prime factors: 53 = 53; 46 = 2 x 23. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 2438. Check: 2438/53 = 46, 2438/46 = 53, so 2438 is exactly divisible by each number.

Let the numbers be 52, 39. Break each into prime factors: 52 = 2 x 2 x 13; 39 = 3 x 13. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 156. Check: 156/52 = 3, 156/39 = 4, so 156 is exactly divisible by each number.

Let the numbers be 52, 40. Break each into prime factors: 52 = 2 x 2 x 13; 40 = 2 x 2 x 2 x 5. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 520. Check: 520/52 = 10, 520/40 = 13, so 520 is exactly divisible by each number.

Let the numbers be 32, 44, 36. Break each into prime factors: 32 = 2 x 2 x 2 x 2 x 2; 44 = 2 x 2 x 11; 36 = 2 x 2 x 3 x 3. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 3168. Check: 3168/32 = 99, 3168/44 = 72, 3168/36 = 88, so 3168 is exactly divisible by each number.

Let the numbers be 19, 32. Break each into prime factors: 19 = 19; 32 = 2 x 2 x 2 x 2 x 2. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 608. Check: 608/19 = 32, 608/32 = 19, so 608 is exactly divisible by each number.

Let the numbers be 45, 35. Break each into prime factors: 45 = 3 x 3 x 5; 35 = 5 x 7. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 315. Check: 315/45 = 7, 315/35 = 9, so 315 is exactly divisible by each number.

Let the numbers be 25, 28, 19. Break each into prime factors: 25 = 5 x 5; 28 = 2 x 2 x 7; 19 = 19. The L.C.M. takes every prime that appears, at its highest power seen. Multiplying those out gives 13300. Check: 13300/25 = 532, 13300/28 = 475, 13300/19 = 700, so 13300 is exactly divisible by each number.

Let the numbers be 96, 236. Break each into prime factors: 96 = 2 x 2 x 2 x 2 x 2 x 3; 236 = 2 x 2 x 59. The H.C.F. is the product of the primes common to every number. That common product is 4. Check: 96/4 = 24, 236/4 = 59, so 4 divides each number exactly.

Let the numbers be 288, 140. Break each into prime factors: 288 = 2 x 2 x 2 x 2 x 2 x 3 x 3; 140 = 2 x 2 x 5 x 7. The H.C.F. is the product of the primes common to every number. That common product is 4. Check: 288/4 = 72, 140/4 = 35, so 4 divides each number exactly.

Let the numbers be 218, 62. Break each into prime factors: 218 = 2 x 109; 62 = 2 x 31. The H.C.F. is the product of the primes common to every number. That common product is 2. Check: 218/2 = 109, 62/2 = 31, so 2 divides each number exactly.

Let the numbers be 250, 286. Break each into prime factors: 250 = 2 x 5 x 5 x 5; 286 = 2 x 11 x 13. The H.C.F. is the product of the primes common to every number. That common product is 2. Check: 250/2 = 125, 286/2 = 143, so 2 divides each number exactly.

Let the numbers be 102, 132. Break each into prime factors: 102 = 2 x 3 x 17; 132 = 2 x 2 x 3 x 11. The H.C.F. is the product of the primes common to every number. That common product is 6. Check: 102/6 = 17, 132/6 = 22, so 6 divides each number exactly.

Let the numbers be 96, 243. Break each into prime factors: 96 = 2 x 2 x 2 x 2 x 2 x 3; 243 = 3 x 3 x 3 x 3 x 3. The H.C.F. is the product of the primes common to every number. That common product is 3. Check: 96/3 = 32, 243/3 = 81, so 3 divides each number exactly.

Let the numbers be 288, 396. Break each into prime factors: 288 = 2 x 2 x 2 x 2 x 2 x 3 x 3; 396 = 2 x 2 x 3 x 3 x 11. The H.C.F. is the product of the primes common to every number. That common product is 36. Check: 288/36 = 8, 396/36 = 11, so 36 divides each number exactly.

Let the numbers be 209, 38. Break each into prime factors: 209 = 11 x 19; 38 = 2 x 19. The H.C.F. is the product of the primes common to every number. That common product is 19. Check: 209/19 = 11, 38/19 = 2, so 19 divides each number exactly.

Let the numbers be 370, 326. Break each into prime factors: 370 = 2 x 5 x 37; 326 = 2 x 163. The H.C.F. is the product of the primes common to every number. That common product is 2. Check: 370/2 = 185, 326/2 = 163, so 2 divides each number exactly.

Let the numbers be 128, 320. Break each into prime factors: 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2; 320 = 2 x 2 x 2 x 2 x 2 x 2 x 5. The H.C.F. is the product of the primes common to every number. That common product is 64. Check: 128/64 = 2, 320/64 = 5, so 64 divides each number exactly.

Let the numbers be 300, 36. Break each into prime factors: 300 = 2 x 2 x 3 x 5 x 5; 36 = 2 x 2 x 3 x 3. The H.C.F. is the product of the primes common to every number. That common product is 12. Check: 300/12 = 25, 36/12 = 3, so 12 divides each number exactly.

Let the numbers be 316, 144, 188. Break each into prime factors: 316 = 2 x 2 x 79; 144 = 2 x 2 x 2 x 2 x 3 x 3; 188 = 2 x 2 x 47. The H.C.F. is the product of the primes common to every number. That common product is 4. Check: 316/4 = 79, 144/4 = 36, 188/4 = 47, so 4 divides each number exactly.

Let the numbers be 377, 260. Break each into prime factors: 377 = 13 x 29; 260 = 2 x 2 x 5 x 13. The H.C.F. is the product of the primes common to every number. That common product is 13. Check: 377/13 = 29, 260/13 = 20, so 13 divides each number exactly.

Let the numbers be 360, 90. Break each into prime factors: 360 = 2 x 2 x 2 x 3 x 3 x 5; 90 = 2 x 3 x 3 x 5. The H.C.F. is the product of the primes common to every number. That common product is 90. Check: 360/90 = 4, 90/90 = 1, so 90 divides each number exactly.