Six fruit baskets contain peaches, apples and pears. The number of peaches in each basket is equal to the total number of apples in the other baskets. The number of apples in each basket is equal to the total number of pears in the other baskets. (a) Find a way to place $31$ fruits in the baskets, satisfying the conditions of the statement. (b) Explain why the total number of fruits must always be multiple of $31$.
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Tags: number theory, algebra
SomeonecoolLovesMaths
24.03.2024 16:14
Basket $1$ has $10$ peaches. Basket $2$ has 10 apples and $1$ pear. Basket $3$ to $6$ have $10$ in total.
BackToSchool
24.03.2024 20:32
parmenides51 wrote: Six fruit baskets contain peaches, apples and pears. The number of peaches in each basket is equal to the total number of apples in the other baskets. The number of apples in each basket is equal to the total number of pears in the other baskets. (a) Find a way to place $31$ fruits in the baskets, satisfying the conditions of the statement. (b) Explain why the total number of fruits must always be multiple of $31$.
Let $a = apple, b = peach, c = pear$.
$$\begin{cases}
\sum_{i=1}^{6} a_i + \sum_{i=1}^{6} b_i + \sum_{i=1}^{6} c_i &= 31 \\
\sum_{i=1}^{6} b_i &= 5 \sum_{i=1}^{6} a_i \\
\sum_{i=1}^{6} a_i &= 5 \sum_{i=1}^{6} c_i
\end{cases} $$$$\implies \sum_{i=1}^{6} a_i = 5, \sum_{i=1}^{6} b_i =25, \sum_{i=1}^{6} c_i =1 $$One possible arrangement is
$$(apple, peaches, pears) = (0, 5, 1), (1, 4, 0), (1, 4, 0), (1, 4, 0), (1, 4, 0), (1, 4, 0)$$
$$\sum_{i=1}^{6} a_i + \sum_{i=1}^{6} b_i + \sum_{i=1}^{6} c_i = 31 \sum_{i=1}^{6} c_i$$Thus, the total number of fruits must be multiple of $31$.