803661374160=2*2*2*2*3*3*3*3*5*17*37*37*73*73
=>2^4*3^4*5^1*17^1*37^2*73*2
hence total no.of factors=5*5*2*2*3*3=900
no.of odd factors=5*2*2*3*3=180
no.of even factors=total-no.of odd factors
=>900-180=720
hence the ans is option(10)

803661374160=2^4*3^4*5^1*17^1*37^2*73^2
=2^a*3^b*5^c*17^d*37^e*73^f
Total factors are=5*5*2*2*3*3=900
In those Odd factors=a=0
A=0
B=0,1,2,3,4
C=0,1
D=0,1
E=0,1,2
F=0,1,2
No of odd factors=1*5*2*2*3*3=180
Even Factors are =900+180=720
My option is 10👈

The possible factors of 803661374160 are,

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 17, 18, 20, 24, 27, 30, 34, 36, 37, 40, 45, 48, 51, 54, 60, 68, 72, 73, 74, 80, 81, 85, 90, 102, 108, 111, 120, 135, 136, 144, 146, 148, 153, 162, 170, 180, 185, 204, 216, 219, 222, 240, 255, 270, 272, 292, 296, 306, 324, 333, 340, 360, 365, 370, 405, 408, 432, 438, 444, 459, 510, 540, 555, 584, 592, 612, 629, 648, 657, 666, 680, 720, 730, 740, 765, 810, 816, 876, 888, 918, 999, 1020, 1080, 1095, 1110, 1168, 1224, 1241, 1258, 1296, 1314, 1332, 1360, 1369, 1377, 1460, 1480, 1530, 1620, 1665, 1752, 1776, 1836, 1887, 1971, 1998, 2040, 2160, 2190, 2220, 2295, 2448, 2482, 2516, 2628, 2664, 2701, 2738, 2754, 2920, 2960, 2997, 3060, 3145, 3240, 3285, 3330, 3504, 3672, 3723, 3774, 3942, 3996, 4080, 4107, 4380, 4440, 4590, 4964, 4995, 5032, 5256, 5328, 5329, 5402, 5476, 5508, 5661, 5840, 5913, 5994, 6120, 6205, 6290, 6480, 6570, 6660, 6845, 6885, 7344, 7446, 7548, 7884, 7992, 8103, 8214, 8760, 8880, 9180, 9435, 9855, 9928, 9990, 10064, 10512, 10658, 10804, 10952, 11016, 11169, 11322, 11826, 11988, 12240, 12321, 12410, 12580, 13140, 13320, 13505, 13690, 13770, 14892, 14985, 15096, 15768, 15984, 15987, 16206, 16428, 16983, 17520, 18360, 18615, 18870, 19710, 19856, 19980, 20535, 21316, 21608, 21904, 22032, 22338, 22644, 23273, 23652, 23976, 24309, 24642, 24820, 25160, 26280, 26640, 26645, 27010, 27380, 27540, 28305, 29565, 29784, 29970, 30192, 31536, 31974, 32412, 32856, 33507, 33966, 36720, 36963, 37230, 37740, 39420, 39960, 40515, 41070, 42632, 43216, 44676, 45288, 45917, 46546, 47304, 47952, 47961, 48618, 49284, 49640, 50320, 50949, 52560, 53290, 54020, 54760, 55080, 55845, 56610, 59130, 59568, 59940, 61605, 63948, 64824, 65712, 67014, 67932, 69819, 72927, 73926, 74460, 75480, 78840, 79920, 79935, 81030, 82140, 84915, 85264, 89352, 90576, 90593, 91834, 93092, 94608, 95922, 97236, 98568, 99280, 99937, 100521, 101898, 106580, 108040, 109520, 110160, 110889, 111690, 113220, 116365, 118260, 119880, 121545, 123210, 127896, 129648, 134028, 135864, 137751, 139638, 143883, 145854, 147852, 148920, 150960, 157680, 159870, 162060, 164280, 167535, 169830, 178704, 181186, 183668, 184815, 186184, 191844, 194472, 197136, 197173, 199874, 201042, 203796, 209457, 213160, 216080, 218781, 221778, 223380, 226440, 229585, 232730, 236520, 239760, 239805, 243090, 246420, 254745, 255792, 268056, 271728, 271779, 275502, 279276, 287766, 291708, 295704, 297840, 299811, 319740, 324120, 328560, 335070, 339660, 349095, 362372, 364635, 367336, 369630, 372368, 383688, 388944, 394346, 399748, 402084, 407592, 413253, 418914, 426320, 431649, 437562, 443556, 446760, 452880, 452965, 459170, 465460, 473040, 479610, 486180, 492840, 499685, 502605, 509490, 536112, 543558, 551004, 554445, 558552, 575532, 583416, 591408, 591519, 599622, 628371, 639480, 648240, 670140, 679320, 688755, 698190, 719415, 724744, 729270, 734672, 739260, 767376, 788692, 799496, 804168, 815184, 815337, 826506, 837828, 863298, 875124, 887112, 893520, 899433, 905930, 918340, 930920, 959220, 972360, 985680, 985865, 999370, 1005210, 1018980, 1047285, 1087116, 1093905, 1102008, 1108890, 1117104, 1151064, 1166832, 1183038, 1199244, 1239759, 1256742, 1278960, 1340280, 1358640, 1358895, 1377510, 1396380, 1438830, 1449488, 1458540, 1478520, 1499055, 1577384, 1598992, 1608336, 1630674, 1653012, 1675656, 1698929, 1726596, 1750248, 1774224, 1774557, 1798866, 1811860, 1836680, 1861840, 1885113, 1918440, 1944720, 1971730, 1998740, 2010420, 2037960, 2066265, 2094570, 2158245, 2174232, 2187810, 2204016, 2217780, 2302128, 2366076, 2398488, 2446011, 2479518, 2513484, 2680560, 2698299, 2717790, 2755020, 2792760, 2877660, 2917080, 2957040, 2957595, 2998110, 3141855, 3154768, 3261348, 3306024, 3351312, 3351941, 3397858, 3453192, 3500496, 3549114, 3597732, 3623720, 3673360, 3719277, 3770226, 3836880, 3943460, 3997480, 4020840, 4075920, 4076685, 4132530, 4189140, 4316490, 4348464, 4375620, 4435560, 4497165, 4732152, 4796976, 4892022, 4959036, 5026968, 5096787, 5323671, 5396598, 5435580, 5510040, 5585520, 5755320, 5834160, 5915190, 5996220, 6198795, 6283710, 6522696, 6612048, 6703882, 6795716, 6906384, 7098228, 7195464, 7247440, 7295401, 7338033, 7438554, 7540452, 7886920, 7994960, 8041680, 8094897, 8153370, 8265060, 8378280, 8494645, 8632980, 8751240, 8871120, 8872785, 8994330, 9425565, 9464304, 9784044, 9918072, 10053936, 10055823, 10193574, 10647342, 10793196, 10871160, 11020080, 11510640, 11830380, 11992440, 12230055, 12397590, 12567420, 13045392, 13407764, 13491495, 13591432, 14196456, 14390928, 14590802, 14676066, 14877108, 15080904, 15290361, 15773840, 15971013, 16189794, 16306740, 16530120, 16756560, 16759705, 16989290, 17265960, 17502480, 17745570, 17988660, 18596385, 18851130, 19568088, 19836144, 20111646, 20387148, 21294684, 21586392, 21742320, 21886203, 23660760, 23984880, 24460110, 24795180, 25134840, 25483935, 26618355, 26815528, 26982990, 27182864, 28392912, 29181604, 29352132, 29754216, 30161808, 30167469, 30580722, 31942026, 32379588, 32613480, 33060240, 33519410, 33978580, 34531920, 35491140, 35977320, 36477005, 36690165, 37192770, 37702260, 39136176, 40223292, 40474485, 40774296, 42589368, 43172784, 43772406, 45871083, 47321520, 48920220, 49590360, 50269680, 50279115, 50967870, 53236710, 53631056, 53965980, 58363208, 58704264, 59508432, 60334938, 61161444, 63884052, 64759176, 65226960, 65658609, 67038820, 67957160, 70982280, 71954640, 72954010, 73380330, 74385540, 75404520, 76451805, 79855065, 80446584, 80948970, 81548592, 85178736, 87544812, 90502407, 91742166, 97840440, 99180720, 100558230, 101935740, 106473420, 107931960, 109431015, 116726416, 117408528, 120669876, 122322888, 124021817, 127768104, 129518352, 131317218, 134077640, 135914320, 137613249, 141964560, 145908020, 146760660, 148771080, 150809040, 150837345, 152903610, 159710130, 160893168, 161897940, 175089624, 181004814, 183484332, 195680880, 196975827, 201116460, 203871480, 212946840, 215863920, 218862030, 229355415, 241339752, 244645776, 248043634, 255536208, 262634436, 268155280, 271507221, 275226498, 291816040, 293521320, 297542160, 301674690, 305807220, 319420260, 323795880, 328293045, 350179248, 362009628, 366968664, 372065451, 393951654, 402232920, 407742960, 425893680, 437724060, 452512035, 458710830, 482679504, 496087268, 525268872, 543014442, 550452996, 583632080, 587042640, 590927481, 603349380, 611614440, 620109085, 638840520, 647591760, 656586090, 688066245, 724019256, 733937328, 744130902, 787903308, 804465840, 875448120, 905024070, 917421660, 984879135, 992174536, 1050537744, 1086028884, 1100905992, 1116196353, 1181854962, 1206698760, 1223228880, 1240218170, 1277681040, 1313172180, 1357536105, 1376132490, 1448038512, 1488261804, 1575806616, 1750896240, 1810048140, 1834843320, 1860327255, 1969758270, 1984349072, 2172057768, 2201811984, 2232392706, 2363709924, 2413397520, 2480436340, 2626344360, 2715072210, 2752264980, 2954637405, 2976523608, 3151613232, 3348589059, 3620096280, 3669686640, 3720654510, 3939516540, 4344115536, 4464785412, 4727419848, 4960872680, 5252688720, 5430144420, 5504529960, 5580981765, 5909274810, 5953047216, 6697178118, 7240192560, 7441309020, 7879033080, 8929570824, 9454839696, 9921745360, 10045767177, 10860288840, 11009059920, 11161963530, 11818549620, 13394356236, 14882618040, 15758066160, 16742945295, 17859141648, 20091534354, 21720577680, 22323927060, 23637099240, 26788712472, 29765236080, 33485890590, 40183068708, 44647854120, 47274198480, 50228835885, 53577424944, 66971781180, 80366137416, 89295708240, 100457671770, 133943562360, 160732274832, 200915343540, 267887124720, 401830687080, 803661374160

Total factor 900,

Even factors,

2, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 34, 36, 40, 48, 54, 60, 68, 72, 74, 80, 90, 102, 108, 120, 136, 144, 146, 148, 162, 170, 180, 204, 216, 222, 240, 270, 272, 292, 296, 306, 324, 340, 360, 370, 408, 432, 438, 444, 510, 540, 584, 592, 612, 648, 666, 680, 720, 730, 740, 810, 816, 876, 888, 918, 1020, 1080, 1110, 1168, 1224, 1258, 1296, 1314, 1332, 1360, 1460, 1480, 1530, 1620, 1752, 1776, 1836, 1998, 2040, 2160, 2190, 2220, 2448, 2482, 2516, 2628, 2664, 2738, 2754, 2920, 2960, 3060, 3240, 3330, 3504, 3672, 3774, 3942, 3996, 4080, 4380, 4440, 4590, 4964, 5032, 5256, 5328, 5402, 5476, 5508, 5840, 5994, 6120, 6290, 6480, 6570, 6660, 7344, 7446, 7548, 7884, 7992, 8214, 8760, 8880, 9180, 9928, 9990, 10064, 10512, 10658, 10804, 10952, 11016, 11322, 11826, 11988, 12240, 12410, 12580, 13140, 13320, 13690, 13770, 14892, 15096, 15768, 15984, 16206, 16428, 17520, 18360, 18870, 19710, 19856, 19980, 21316, 21608, 21904, 22032, 22338, 22644, 23652, 23976, 24642, 24820, 25160, 26280, 26640, 27010, 27380, 27540, 29784, 29970, 30192, 31536, 31974, 32412, 32856, 33966, 36720, 37230, 37740, 39420, 39960, 41070, 42632, 43216, 44676, 45288, 46546, 47304, 47952, 48618, 49284, 49640, 50320, 52560, 53290, 54020, 54760, 55080, 56610, 59130, 59568, 59940, 63948, 64824, 65712, 67014, 67932, 73926, 74460, 75480, 78840, 79920, 81030, 82140, 85264, 89352, 90576, 91834, 93092, 94608, 95922, 97236, 98568, 99280, 101898, 106580, 108040, 109520, 110160, 111690, 113220, 118260, 119880, 123210, 127896, 129648, 134028, 135864, 139638, 145854, 147852, 148920, 150960, 157680, 159870, 162060, 164280, 169830, 178704, 181186, 183668, 186184, 191844, 194472, 197136, 199874, 201042, 203796, 213160, 216080, 221778, 223380, 226440, 232730, 236520, 239760, 243090, 246420, 255792, 268056, 271728, 275502, 279276, 287766, 291708, 295704, 297840, 319740, 324120, 328560, 335070, 339660, 362372, 367336, 369630, 372368, 383688, 388944, 394346, 399748, 402084, 407592, 418914, 426320, 437562, 443556, 446760, 452880, 459170, 465460, 473040, 479610, 486180, 492840, 509490, 536112, 543558, 551004, 558552, 575532, 583416, 591408, 599622, 639480, 648240, 670140, 679320, 698190, 724744, 729270, 734672, 739260, 767376, 788692, 799496, 804168, 815184, 826506, 837828, 863298, 875124, 887112, 893520, 905930, 918340, 930920, 959220, 972360, 985680, 999370, 1005210, 1018980, 1087116, 1102008, 1108890, 1117104, 1151064, 1166832, 1183038, 1199244, 1256742, 1278960, 1340280, 1358640, 1377510, 1396380, 1438830, 1449488, 1458540, 1478520, 1577384, 1598992, 1608336, 1630674, 1653012, 1675656, 1726596, 1750248, 1774224, 1798866, 1811860, 1836680, 1861840, 1918440, 1944720, 1971730, 1998740, 2010420, 2037960, 2094570, 2174232, 2187810, 2204016, 2217780, 2302128, 2366076, 2398488, 2479518, 2513484, 2680560, 2717790, 2755020, 2792760, 2877660, 2917080, 2957040, 2998110, 3154768, 3261348, 3306024, 3351312, 3397858, 3453192, 3500496, 3549114, 3597732, 3623720, 3673360, 3770226, 3836880, 3943460, 3997480, 4020840, 4075920, 4132530, 4189140, 4316490, 4348464, 4375620, 4435560, 4732152, 4796976, 4892022, 4959036, 5026968, 5396598, 5435580, 5510040, 5585520, 5755320, 5834160, 5915190, 5996220, 6283710, 6522696, 6612048, 6703882, 6795716, 6906384, 7098228, 7195464, 7247440, 7438554, 7540452, 7886920, 7994960, 8041680, 8153370, 8265060, 8378280, 8632980, 8751240, 8871120, 8994330, 9464304, 9784044, 9918072, 10053936, 10193574, 10647342, 10793196, 10871160, 11020080, 11510640, 11830380, 11992440, 12397590, 12567420, 13045392, 13407764, 13591432, 14196456, 14390928, 14590802, 14676066, 14877108, 15080904, 15773840, 16189794, 16306740, 16530120, 16756560, 16989290, 17265960, 17502480, 17745570, 17988660, 18851130, 19568088, 19836144, 20111646, 20387148, 21294684, 21586392, 21742320, 23660760, 23984880, 24460110, 24795180, 25134840, 26815528, 26982990, 27182864, 28392912, 29181604, 29352132, 29754216, 30161808, 30580722, 31942026, 32379588, 32613480, 33060240, 33519410, 33978580, 34531920, 35491140, 35977320, 37192770, 37702260, 39136176, 40223292, 40774296, 42589368, 43172784, 43772406, 47321520, 48920220, 49590360, 50269680, 50967870, 53236710, 53631056, 53965980, 58363208, 58704264, 59508432, 60334938, 61161444, 63884052, 64759176, 65226960, 67038820, 67957160, 70982280, 71954640, 72954010, 73380330, 74385540, 75404520, 80446584, 80948970, 81548592, 85178736, 87544812, 91742166, 97840440, 99180720, 100558230, 101935740, 106473420, 107931960, 116726416, 117408528, 120669876, 122322888, 127768104, 129518352, 131317218, 134077640, 135914320, 141964560, 145908020, 146760660, 148771080, 150809040, 152903610, 159710130, 160893168, 161897940, 175089624, 181004814, 183484332, 195680880, 201116460, 203871480, 212946840, 215863920, 218862030, 241339752, 244645776, 248043634, 255536208, 262634436, 268155280, 275226498, 291816040, 293521320, 297542160, 301674690, 305807220, 319420260, 323795880, 350179248, 362009628, 366968664, 393951654, 402232920, 407742960, 425893680, 437724060, 458710830, 482679504, 496087268, 525268872, 543014442, 550452996, 583632080, 587042640, 603349380, 611614440, 638840520, 647591760, 656586090, 724019256, 733937328, 744130902, 787903308, 804465840, 875448120, 905024070, 917421660, 992174536, 1050537744, 1086028884, 1100905992, 1181854962, 1206698760, 1223228880, 1240218170, 1277681040, 1313172180, 1376132490, 1448038512, 1488261804, 1575806616, 1750896240, 1810048140, 1834843320, 1969758270, 1984349072, 2172057768, 2201811984, 2232392706, 2363709924, 2413397520, 2480436340, 2626344360, 2715072210, 2752264980, 2976523608, 3151613232, 3620096280, 3669686640, 3720654510, 3939516540, 4344115536, 4464785412, 4727419848, 4960872680, 5252688720, 5430144420, 5504529960, 5909274810, 5953047216, 6697178118, 7240192560, 7441309020, 7879033080, 8929570824, 9454839696, 9921745360, 10860288840, 11009059920, 11161963530, 11818549620, 13394356236, 14882618040, 15758066160, 17859141648, 20091534354, 21720577680, 22323927060, 23637099240, 26788712472, 29765236080, 33485890590, 40183068708, 44647854120, 47274198480, 53577424944, 66971781180, 80366137416, 89295708240, 100457671770, 133943562360, 160732274832, 200915343540, 267887124720, 401830687080, 803661374160

Total number even factors 720,

The answer is not available on the option, hence option 10 is correct.

803661374160 = (2^4)*(3^4)*5*17*(37^2)*(73^2)
=> Total number of factors = (4+1) * (4+1) * (1+1) * (1+1) * (2+1) * (2+1)
= 5*5*2*2*3*3
= 900

Number of odd factors = (4+1) * (1+) * (1+) * (2*1) * (2*1)
= 180

So,number of even factors = 900-180 = 720