Question

po There is a group of soldiers. When they are lined in rows of 6, 8 or 15, there are 5 soldiers left. (a) Find the minimum number of soldiers in the (b) group. If the group can be exactly divided into small teams of 13 soldiers each, find the minimum number of soldiers in the group.​

Answer 1

The minimum number of soldiers in the group is 125 if it cannot be divided into teams of 13 soldiers each, and 1000 if it can be.

Step-by-step explanation:

To find the minimum number of soldiers in the group, we need to find the least common multiple (LCM) of 6, 8, and 15. The LCM is the smallest number that is divisible by each of these numbers.

1. Start by finding the prime factorization of each number: 6 = 2 * 3, 8 = 2^3, and 15 = 3 * 5.
2. Take the highest power of each prime factor: 2^3 * 3 * 5 = 120.
3. Add 5 to get the minimum number of soldiers: 120 + 5 = 125 soldiers.

To find the minimum number of soldiers in the group if it can be divided exactly into small teams of 13 soldiers each, we need to find the smallest multiple of 13 that is also a multiple of the minimum number of soldiers calculated earlier (125).

1. Start by finding the multiples of 13: 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208, 221, 234, 247, 260, 273, 286, 299, 312, 325, 338, 351, 364, 377, 390, 403, 416, 429, 442, 455, 468, 481, 494, 507, 520, 533, 546, 559, 572, 585, 598, 611, 624, 637, 650, 663, 676, 689, 702, 715, 728, 741, 754, 767, 780, 793, 806, 819, 832, 845, 858, 871, 884, 897, 910, 923, 936, 949, 962, 975, 988, 1001.
2. Find the smallest multiple that is divisible by 125: 1000.

Therefore, the minimum number of soldiers in the group is 125 if it cannot be divided into teams of 13 soldiers each and 1000 if it can be.

Learn more about minimum number of soldiers in a group