There are large number of silver coins weighing 2 gm, 5 gm, 10 gm, 25 gm, 50 gm each. Consider the following statements:
1. To buy 78 gm of coins one must buy at least 7 coins.
2. To weigh 78 gm using these coins one can use less than 7 coins.

Which of the statements given above is/are correct?

Correct Answer: (C) Both 1 and 2

Coins are available in denominations of 2, 5, 10, 25, and 50 gm. In Statement 1, purchasing 78 gm requires non-negative linear combinations of coin weights. Using the largest coin 50 gm leaves 28 gm; to make 28 gm using denominations $\le 25$$\le$ 25, the minimal coin count is two 10 gm coins and four 2 gm coins ($10 + 10 + 2 + 2 + 2 + 2 = 28$), totaling $1 + 6 = 7$ coins. Any other combination requires at least 7 coins (Statement 1 is correct). In Statement 2, weighing an object on a two-pan balance allows placing coins on both pans. Placing 50 gm, 25 gm, and 5 gm coins on one pan (80 gm) and a 2 gm coin with the 78 gm object on the other pan ($78 + 2 = 80$$\text{ gm}$) achieves balance using only 4 coins, which is strictly less than 7 (Statement 2 is correct).