Question

Фокусник задумал два натуральных числа и сообщил Симе их сумму, а Прову — их произведение. Зная, что произведение равно 2280, Пров смог отгадать задуманные числа только после того, как Сима сообщила, что сумма у неё нечётна и двузначна. Так какие числа задумал фокусник?

Answer 1

Ответ:

57 40

Объяснение:

2280=2^3*3*5*19

1 число-3*19=57

2 число-5*2^3=40

Answer 2

Problem Analysis

In this problem, a magician has thought of two natural numbers and shared their sum with Sime and their product with Pro. Pro is able to determine the two numbers after Sime reveals that the sum is odd and two-digit. We need to find the two numbers that the magician thought of.

Solution

Let's denote the two numbers as x and y. We know that the product of x and y is 2280. Additionally, we know that the sum of x and y is odd and two-digit.

To find the two numbers, we can start by finding the factors of 2280. By finding the factors, we can check which pairs of numbers satisfy the condition of having an odd and two-digit sum.

Let's calculate the factors of 2280:

2280 = 2 * 2 * 2 * 3 * 5 * 19

Now, let's consider the pairs of factors that have an odd and two-digit sum:

- Pair 1: (2, 1140) -> sum = 2 + 1140 = 1142 (not odd) - Pair 2: (4, 570) -> sum = 4 + 570 = 574 (odd and two-digit) - Pair 3: (8, 285) -> sum = 8 + 285 = 293 (odd and two-digit) - Pair 4: (3, 760) -> sum = 3 + 760 = 763 (odd and two-digit) - Pair 5: (5, 456) -> sum = 5 + 456 = 461 (odd and two-digit) - Pair 6: (6, 380) -> sum = 6 + 380 = 386 (even) - Pair 7: (10, 228) -> sum = 10 + 228 = 238 (even) - Pair 8: (12, 190) -> sum = 12 + 190 = 202 (even) - Pair 9: (15, 152) -> sum = 15 + 152 = 167 (odd and two-digit) - Pair 10: (19, 120) -> sum = 19 + 120 = 139 (odd and two-digit) - Pair 11: (20, 114) -> sum = 20 + 114 = 134 (even) - Pair 12: (30, 76) -> sum = 30 + 76 = 106 (even) - Pair 13: (38, 60) -> sum = 38 + 60 = 98 (even) - Pair 14: (57, 40) -> sum = 57 + 40 = 97 (odd and two-digit) - Pair 15: (76, 30) -> sum = 76 + 30 = 106 (even) - Pair 16: (114, 20) -> sum = 114 + 20 = 134 (even) - Pair 17: (120, 19) -> sum = 120 + 19 = 139 (odd and two-digit) - Pair 18: (152, 15) -> sum = 152 + 15 = 167 (odd and two-digit) - Pair 19: (190, 12) -> sum = 190 + 12 = 202 (even) - Pair 20: (228, 10) -> sum = 228 + 10 = 238 (even) - Pair 21: (285, 8) -> sum = 285 + 8 = 293 (odd and two-digit) - Pair 22: (380, 6) -> sum = 380 + 6 = 386 (even) - Pair 23: (456, 5) -> sum = 456 + 5 = 461 (odd and two-digit) - Pair 24: (570, 4) -> sum = 570 + 4 = 574 (odd and two-digit) - Pair 25: (1140, 2) -> sum = 1140 + 2 = 1142 (not odd)

From the pairs above, we can see that there are three pairs that satisfy the condition of having an odd and two-digit sum: (4, 570), (3, 760), and (5, 456). Therefore, the magician could have thought of any of these three pairs of numbers.

To summarize, the possible pairs of numbers that the magician could have thought of are: - (4, 570) - (3, 760) - (5, 456)

Please note that there may be other pairs of numbers that satisfy the given conditions, but these are the ones we found based on the factors of 2280.

I hope this helps! Let me know if you have any further questions.