In how many ways can you express the number 2003 as a sum of two prime numbers?
A) 1
B) 2
C) 3
D) 4
B) 2
To find the number of ways to express the number 2003 as a sum of two prime numbers, we need to check all possible combinations of prime numbers.
First, let's determine all the prime numbers that are less than or equal to 2003:
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, 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.
Now, let's check all the possible combinations:
2 + 2001 = 2003 (Prime numbers: 2, 2003)
7 + 1996 = 2003 (Prime numbers: 7, 1996)
19 + 1984 = 2003 (Prime numbers: 19, 1984)
61 + 1942 = 2003 (Prime numbers: 61, 1942)
137 + 1866 = 2003 (Prime numbers: 137, 1866)
577 + 1426 = 2003 (Prime numbers: 577, 1426)
805 + 1198 = 2003 (Prime numbers: 805, 1198)
1061 + 942 = 2003 (Prime numbers: 1061, 942)
...
By checking all possible combinations, we can see that there are two ways to express the number 2003 as a sum of two prime numbers.
Therefore, the answer is B) 2.
To find the number of ways to express 2003 as a sum of two prime numbers, you can follow these steps:
1. Start by listing all prime numbers up to the square root of 2003. The largest prime number less than or equal to the square root of 2003 is 43 since 43 * 43 = 1849.
2. Next, subtract each prime number from 2003 and check if the result is prime as well. If it is, then you have found a valid pair.
Let's go through the process:
List of primes up to the square root of 2003: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43
Now, subtract each prime number from 2003 and check if the result is prime:
2003 - 2 = 2001 (not prime)
2003 - 3 = 2000 (not prime)
2003 - 5 = 1998 (not prime)
2003 - 7 = 1996 (not prime)
2003 - 11 = 1992 (not prime)
2003 - 13 = 1990 (not prime)
2003 - 17 = 1986 (not prime)
2003 - 19 = 1984 (not prime)
2003 - 23 = 1980 (not prime)
2003 - 29 = 1974 (not prime)
2003 - 31 = 1972 (not prime)
2003 - 37 = 1966 (not prime)
2003 - 41 = 1962 (not prime)
2003 - 43 = 1960 (not prime)
Since none of the differences are prime, there are no valid pairs that sum up to 2003. Therefore, the answer is A) 1.