Question

44 балла: тема : НОД,НОК Найдите наименьшее число , которое при делении на 7, 8, 9, 10 в остатке дает 2.

Answer 1

Найдем НОК (7,8,9,10) = НОК(7,8,9,5)=7*8*9*5=2520 - Оно делится на 7,8,9,10 
Тогда наименьшее число , которое при делении на 7, 8, 9, 10 в остатке дает 2 будет 2522

Answer 2

Finding the Smallest Number Divisible by 7, 8, 9, and 10 with a Remainder of 2

To find the smallest number that, when divided by 7, 8, 9, and 10, leaves a remainder of 2, we can use the concept of the least common multiple (LCM) of these numbers.

The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set. Therefore, to find the smallest number that satisfies the given conditions, we need to find the LCM of 7, 8, 9, and 10 and then add 2 to the result.

Calculating the LCM

To calculate the LCM of 7, 8, 9, and 10, we can use the following steps:

1. Prime Factorization: Express each number as a product of its prime factors. 2. Select the Highest Power: For each prime factor, select the highest power to form the LCM.

Let's calculate the LCM step by step:

1. Prime Factorization: - 7: 7 is a prime number, so its prime factorization is 7. - 8: 8 = 2^3. - 9: 9 = 3^2. - 10: 10 = 2 * 5.

2. Select the Highest Power: - LCM(7, 8, 9, 10) = 2^3 * 3^2 * 5 * 7 = 2520.

Finding the Smallest Number

Now that we have the LCM, we need to find the smallest number that, when divided by 7, 8, 9, and 10, leaves a remainder of 2. We can do this by adding 2 to the LCM:

Smallest Number = LCM + 2 = 2520 + 2 = 2522.

Therefore, the smallest number that, when divided by 7, 8, 9, and 10, leaves a remainder of 2 is 2522.

Conclusion

The smallest number that satisfies the given conditions is 2522.