According to a regulation like a figure below, coins are arranged as the 1st Fig., 2nd Fig., 3rd Fig. ------ .
Answer the following questions.
(1) Find the number of the coin arranged in the 15th Fig.
(2) How many times are there that the total number of the coin becomes a multiple of 90 from the 1st Fig. to the 50th Fig.?
Answer
(1) 1800 pieces
(2) 3 times
Solution
(1) 8 times of the number of coins in a red square of the figure below is the total number of coins arranged in one Fig.
The number of coins in the red square is increased as 1 × 1, 2 × 2, 3 × 3 ----.
Therefore the number of the coin of the 15th figure is 15 × 15 × 8 = 1800.
(2) 90 = 2 × 3 × 3 ×5 and 8 is a multiple of 2.
The number of coins in the square of Ath Fig. is A × A.
It's necessary that A × A becomes a multiple of 45 for the number of coins in the Fig. to become a multiple of 90.
Because it's 45 = 3 × 3 × 5, A has 3 and 5 as divisors and is a multiple of 15.
Therefore it's three times of the 15th Fig., the 30th and the 45th that the total number of coins in Fig. becomes a multiple of 90.
