Every 12th customer at a Chicago deli gets a free soda. Every 15th customer gets a free bagel. Which is the first customer to get a free bagel and a free soda?

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In order for the customer to get a free soda n = 12k for some k and for the same customer n = 15j for some j. So, we must have 12k = 15j. But 12 = 2 * 2 * 3 and 15 = 3 *5, so we need k and k so that

2 * 2 * 3 * k = 3 * 5 * j. Dividing both side by 3, this means we need 4k = 5j. If k = 5 and j = 4 we have 12 * k = 60 and 15 * j = 60, so on day 60, the customer gets a free bagel and a free soda.

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