Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.4 AIME, harder Prove it Russia

In a product of 5 positive integers each multiple was decreased by 3. Could it happen that the product becomes exactly 15 times larger than the initial one?

В произведении 5 положительных целых чисел каждый сомножитель уменьшили на 3. Могло ли получиться так, что произведение увеличилось ровно в 15 раз?

Solutions — 2

Solution 1

Yes, it is possible.

An example is $1 \cdot 1 \cdot 1 \cdot 1 \cdot 48$ .

After decreasing each factor by $3$ , we get:
$(-2) \cdot (-2) \cdot (-2) \cdot (-2) \cdot 45 = 16 \cdot 45 = 720.$
The original product is $1 \cdot 1 \cdot 1 \cdot 1 \cdot 48 = 48$ .

Thus, $720 = 15 \cdot 48$ .

Solution 2

Да, могло.

В качестве примера подходит произведение $1 \cdot 1 \cdot 1 \cdot 1 \cdot 48$ .
После указанной операции получается
$(-2) \cdot (-2) \cdot (-2) \cdot (-2) \cdot 45 = 720 = 15 \cdot 48.$

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