A person X wants to distribute some pens among six children A, B, C, D, E, and F. Suppose A gets twice the number of pens received by B, three times that of C, four times that of D, five times that of E and six times that of F. What is the minimum number of pens X should buy so that the number of pens each one gets is an even number?

Let the number of pens received by children A, B, C, D, E, and F be a, b, c, d, e, f respectively. From the given conditions: a = 2b => b = a/2 a = 3c => c = a/3 a = 4d => d = a/4 a = 5e => e = a/5 a = 6f => f = a/6 For a, b, c, d, e, f to be whole numbers, 'a' must be a multiple of 2, 3, 4, 5, and 6. The least common multiple (LCM) of (2, 3, 4, 5, 6) is 60. So, 'a' must be a multiple of 60. Let a = 60k, where k is a positive integer. Now, let's find the number of pens for each child in terms of k: a = 60k b = 60k / 2 = 30k c = 60k / 3 = 20k d = 60k / 4 = 15k e = 60k / 5 = 12k f = 60k / 6 = 10k The total number of pens X should buy is the sum of pens for all children: Total = a + b + c + d + e + f Total = 60k + 30k + 20k + 15k + 12k + 10k Total = 147k Now, consider the additional condition: "the number of pens each one gets is an even number". Let's check each child's pens: a = 60k (Always even, since 60 is even) b = 30k (Always even, since 30 is even) c = 20k (Always even, since 20 is even) d = 15k (For 15k to be an even number, k must be an even number.) e = 12k (Always even, since 12 is even) f = 10k (Always even, since 10 is even) The crucial condition is that 'd' (15k) must be even. This implies that 'k' must be an even number. To find the minimum number of pens, we need the minimum positive integer value for k that is even. The minimum even positive integer for k is 2. Substitute k = 2 into the total number of pens: Total = 147 * 2 = 294 Let's verify the number of pens for each child with k=2: a = 60 * 2 = 120 (even) b = 30 * 2 = 60 (even) c = 20 * 2 = 40 (even) d = 15 * 2 = 30 (even) e = 12 * 2 = 24 (even) f = 10 * 2 = 20 (even) All conditions are met. Therefore, the minimum number of pens X should buy is 294. Analyzing the options: A) 147: This would be the total if k=1. However, if k=1, d=15, which is not an even number. So, this is incorrect. B) 150: Not a multiple of 147. Incorrect. C) 294: This is the calculated minimum number when k=2, satisfying all conditions. Correct. D) 300: Not a multiple of 147. Incorrect. The final answer is C.