#!/usr/bin/env python from math import * def great(m,n): #finds gcd of m,n>0 while n>0: m,n=n,m%n return m def sieve(n): #Sieve of Erthostenes, gives primes up to n. candidates=range(3,n,2) for p in candidates: if p: if p*p>n: break for q in xrange(p*p,n,2*p): candidates[(q-3)/2]=0 return [2]+filter(None, candidates) primelist=[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973] #primitiva r\"otter: [[3, 2], [5, 2], [7, 3], [11, 2], [13, 2], [17, 3], [19, 2], [23, 5], [29, 2], [31, 3], [37, 2], [41, 6], [43, 3], [47, 5], [53, 2], [59,2], [61, 2], [67, 2], [71, 7], [73, 5], [79, 3], [83, 2], [89, 3], [97, 5], [101, 2], [103, 5], [107, 2], [109, 6], [113, 3], [127, 3], [131, 2], [137, 3], [139, 2], [149, 2], [151, 6], [157, 5], [163, 2], [167, 5], [173, 2], [179, 2], [181, 2], [191, 19], [193, 5], [197, 2], [199, 3], [211, 2], [223, 3], [227, 2], [229, 6], [233, 3], [239, 7], [241, 7], [251, 6], [257, 3], [263, 5], [269, 2], [271, 6], [277, 5], [281, 3], [283, 3], [293, 2], [307, 5], [311, 17], [313, 10], [317,2], [331, 3], [337, 10], [347, 2], [349, 2], [353, 3], [359, 7], [367,6], [373, 2], [379, 2], [383, 5], [389, 2], [397, 5], [401, 3], [409,21], [419, 2], [421, 2], [431, 7], [433, 5], [439, 15], [443, 2], [449, 3], [457, 13], [461, 2], [463, 3], [467, 2], [479, 13], [487,3], [491, 2], [499, 7], [503, 5], [509, 2], [521, 3], [523, 2], [541,2]] [[547, 2], [557, 2], [563, 2], [569, 3], [571, 3], [577, 5], [587, 2], [593, 3], [599, 7], [601, 7], [607, 3], [613, 2], [617, 3], [619, 2], [631, 3], [641, 3], [643, 11], [647, 5], [653, 2], [659, 2], [661, 2], [673, 5], [677, 2], [683, 5], [691, 3], [701, 2], [709, 2], [719, 11], [727, 5], [733, 6], [739, 3], [743, 5], [751, 3], [757, 2], [761, 6], [769, 11], [773, 2], [787, 2], [797, 2], [809, 3], [811, 3], [821, 2], [823, 3], [827, 2], [829, 2], [839, 11], [853, 2], [857, 3], [859, 2], [863, 5], [877, 2], [881, 3], [883, 2], [887, 5], [907, 2], [911, 17], [919, 7], [929, 3], [937, 5], [941, 2], [947, 2], [953, 3], [967, 5], [971, 6], [977, 3], [983, 5], [991, 6], [997, 7], [1009, 11], [1013,3], [1019, 2], [1021, 10], [1031, 14], [1033, 5], [1039, 3], [1049,3], [1051, 7], [1061, 2], [1063, 3], [1069, 6], [1087, 3], [1091, 2], [1093, 5], [1097, 3], [1103, 5], [1109, 2], [1117, 2], [1123, 2], [1129, 11], [1151, 17], [1153, 5], [1163, 5], [1171, 2], [1181, 7], [1187, 2], [1193, 3], [1201, 11], [1213, 2], [1217, 3], [1223, 5], [1229, 2], [1231, 3], [1237, 2], [1249, 7], [1259, 2], [1277, 2], [1279, 3], [1283, 2], [1289, 6], [1291, 2], [1297, 10], [1301, 2], [1303, 6], [1307, 2], [1319, 13], [1321, 13], [1327, 3], [1361, 3], [1367, 5], [1373, 2], [1381, 2], [1399, 13], [1409, 3], [1423, 3], [1427, 2], [1429, 6], [1433, 3], [1439, 7], [1447, 3], [1451, 2], [1453, 2], [1459, 3], [1471, 6], [1481, 3], [1483, 2], [1487, 5], [1489, 14], [1493, 2], [1499, 2], [1511, 11], [1523, 2], [1531, 2], [1543, 5], [1549, 2], [1553, 3], [1559, 19], [1567, 3], [1571, 2], [1579, 3], [1583, 5], [1597, 11], [1601, 3], [1607, 5], [1609, 7], [1613, 3], [1619, 2], [1621, 2], [1627, 3], [1637, 2], [1657, 11], [1663, 3], [1667, 2], [1669, 2], [1693, 2], [1697, 3], [1699, 3], [1709, 3], [1721, 3], [1723, 3], [1733, 2], [1741, 2], [1747, 2], [1753, 7], [1759, 6], [1777, 5], [1783, 10], [1787, 2], [1789, 6], [1801, 11], [1811, 6], [1823, 5], [1831, 3], [1847, 5], [1861, 2], [1867, 2], [1871, 14], [1873, 10], [1877, 2], [1879, 6], [1889, 3], [1901, 2], [1907, 2], [1913, 3], [1931, 2], [1933, 5], [1949, 2], [1951, 3], [1973, 2], [1979, 2], [1987, 2]] [[1993, 5], [1997, 2], [1999, 3], [2003, 5], [2011, 3], [2017, 5], [2027, 2], [2029, 2], [2039, 7], [2053, 2], [2063, 5], [2069, 2], [2081, 3], [2083, 2], [2087, 5], [2089, 7], [2099, 2], [2111, 7], [2113, 5], [2129, 3], [2131, 2], [2137, 10], [2141, 2], [2143, 3], [2153, 3], [2161, 23], [2179, 7], [2203, 5], [2207, 5], [2213, 2], [2221, 2], [2237, 2], [2239, 3], [2243, 2], [2251, 7], [2267, 2], [2269, 2], [2273, 3], [2281, 7], [2287, 19], [2293, 2], [2297, 5], [2309, 2], [2311, 3], [2333, 2], [2339, 2], [2341, 7], [2347, 3], [2351, 13], [2357, 2], [2371, 2], [2377, 5], [2381, 3], [2383, 5], [2389, 2], [2393, 3], [2399, 11], [2411, 6], [2417, 3], [2423, 5], [2437, 2], [2441, 6], [2447, 5], [2459, 2], [2467, 2], [2473, 5], [2477, 2], [2503, 3], [2521, 17], [2531, 2], [2539, 2], [2543, 5], [2549, 2], [2551, 6], [2557, 2], [2579, 2], [2591, 7], [2593, 7], [2609, 3], [2617, 5], [2621, 2], [2633, 3], [2647, 3], [2657, 3], [2659, 2], [2663, 5], [2671, 7], [2677, 2], [2683, 2], [2687, 5], [2689, 19], [2693, 2], [2699, 2], [2707, 2], [2711, 7], [2713, 5], [2719, 3], [2729, 3], [2731, 3], [2741, 2], [2749, 6], [2753, 3], [2767, 3], [2777, 3], [2789, 2], [2791, 6], [2797, 2], [2801, 3], [2803, 2], [2819, 2], [2833, 5], [2837, 2], [2843, 2], [2851, 2], [2857, 11], [2861, 2], [2879, 7], [2887, 5], [2897, 3], [2903, 5], [2909, 2], [2917, 5], [2927, 5], [2939, 2], [2953, 13], [2957, 2], [2963, 2], [2969, 3], [2971, 10], [2999, 17], [3001, 14], [3011, 2], [3019, 2], [3023, 5], [3037, 2], [3041, 3], [3049, 11], [3061, 6], [3067, 2], [3079, 6], [3083, 2], [3089, 3], [3109, 6], [3119, 7], [3121, 7], [3137, 3], [3163, 3], [3167, 5], [3169, 7], [3181, 7], [3187, 2], [3191, 11], [3203, 2], [3209, 3], [3217, 5], [3221, 10], [3229, 6], [3251, 6], [3253, 2], [3257, 3], [3259, 3], [3271, 3], [3299, 2], [3301, 6], [3307, 2], [3313, 10], [3319, 6], [3323, 2], [3329, 3], [3331, 3], [3343, 5], [3347, 2], [3359, 11], [3361, 22], [3371, 2], [3373, 5], [3389, 3], [3391, 3], [3407, 5], [3413, 2], [3433, 5], [3449, 3], [3457, 7], [3461, 2], [3463, 3], [3467, 2], [3469, 2], [3491, 2], [3499, 2], [3511, 7], [3517, 2], [3527, 5], [3529, 17], [3533, 2], [3539, 2], [3541, 7], [3547, 2], [3557, 2], [3559, 3], [3571, 2]] [[3581, 2], [3583, 3], [3593, 3], [3607, 5], [3613, 2], [3617, 3], [3623, 5], [3631, 15], [3637, 2], [3643, 2], [3659, 2], [3671, 13], [3673, 5], [3677, 2], [3691, 2], [3697, 5], [3701, 2], [3709, 2], [3719, 7], [3727, 3], [3733, 2], [3739, 7], [3761, 3], [3767, 5], [3769, 7], [3779, 2], [3793, 5], [3797, 2], [3803, 2], [3821, 3], [3823, 3], [3833, 3], [3847, 5], [3851, 2], [3853, 2], [3863, 5], [3877, 2], [3881, 13], [3889, 11], [3907, 2], [3911, 13], [3917, 2], [3919, 3], [3923, 2], [3929, 3], [3931, 2], [3943, 3], [3947, 2], [3967, 6], [3989, 2], [4001, 3], [4003, 2], [4007, 5], [4013, 2], [4019, 2], [4021, 2], [4027, 3], [4049, 3], [4051, 10], [4057, 5], [4073, 3], [4079, 11], [4091, 2], [4093, 2], [4099, 2], [4111, 12], [4127, 5], [4129, 13], [4133, 2], [4139, 2], [4153, 5], [4157, 2], [4159, 3], [4177, 5], [4201, 11], [4211, 6], [4217, 3], [4219, 2], [4229, 2], [4231, 3], [4241, 3], [4243, 2], [4253, 2], [4259, 2], [4261, 2], [4271, 7], [4273, 5], [4283, 2], [4289, 3], [4297, 5], [4327, 3], [4337, 3], [4339, 10], [4349, 2], [4357, 2], [4363, 2], [4373, 2], [4391, 14], [4397, 2], [4409, 3], [4421, 3], [4423, 3], [4441, 21], [4447, 3], [4451, 2], [4457, 3], [4463, 5], [4481, 3], [4483, 2], [4493, 2], [4507, 2], [4513, 7], [4517, 2], [4519, 3], [4523, 5], [4547, 2], [4549, 6], [4561, 11], [4567, 3], [4583, 5], [4591, 11], [4597, 5], [4603, 2], [4621, 2], [4637, 2], [4639, 3], [4643, 5], [4649, 3], [4651, 3], [4657, 15], [4663, 3], [4673, 3], [4679, 11], [4691, 2], [4703, 5], [4721, 6], [4723, 2], [4729, 17], [4733, 5], [4751, 19], [4759, 3], [4783, 6], [4787, 2], [4789, 2], [4793, 3], [4799, 7], [4801, 7], [4813, 2], [4817, 3], [4831, 3], [4861, 11], [4871, 11], [4877, 2], [4889, 3], [4903, 3], [4909, 6], [4919, 13], [4931, 6], [4933, 2], [4937, 3], [4943, 7], [4951, 6], [4957, 2], [4967, 5], [4969, 11], [4973, 2], [4987, 2], [4993, 5], [4999, 3], [5003, 2], [5009, 3], [5011, 2], [5021, 3], [5023, 3], [5039, 11], [5051, 2], [5059, 2], [5077, 2], [5081, 3], [5087, 5], [5099, 2], [5101, 6], [5107, 2], [5113, 19], [5119, 3], [5147, 2], [5153, 5], [5167, 6], [5171, 2], [5179, 2], [5189, 2], [5197, 7], [5209, 17], [5227, 2], [5231, 7], [5233, 10], [5237, 3], [5261, 2], [5273, 3], [5279, 7]]