Question

1). Знайти НСД і НСК чисел а i b, якщо а = 2·5-7-11 i b = 2-3-7. 2). Які з чисел 30; 21; 405; 972; 865 діляться : а) на 9; б) на 5 ? 3). Розкласти на прості множники число 660. 4) Яку найбільшу кількість однакових наборів можна скласти, маючи 24 олівці і 32 ручки так, щоб використати всі олівці і всі ручки та щоб кожен набір містив і олівці, і ручки? 5) Жінка принесла для продажу на ринку певну кількість яблук, більшу за 273, але меншу від 389. Відомо, що ці яблука можна порівну розкласти на 6; 7 або 8 купок. Скільки яблук жінка принесла на ринок?​

Answer 1

Ответ:

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Answer 2

1) To find the greatest common divisor (GCD) and least common multiple (LCM) of two numbers a and b, we can use the following formulas:

GCD(a, b) = GCD(b, a % b) LCM(a, b) = (a * b) / GCD(a, b)

Let's calculate the GCD and LCM for the given numbers:

a = 2 * 5 - 7 - 11 = -11 b = 2 - 3 - 7 = -8

GCD(-11, -8) = GCD(-8, -11 % -8) = GCD(-8, -3) = GCD(-3, -8 % -3) = GCD(-3, -2) = GCD(-2, -3 % -2) = GCD(-2, -1) = GCD(-1, -2 % -1) = GCD(-1, 0) = 1

LCM(-11, -8) = (-11 * -8) / GCD(-11, -8) = 88 / 1 = 88

Therefore, the GCD of -11 and -8 is 1, and the LCM is 88.

2) Let's check which of the numbers 30, 21, 405, 972, and 865 are divisible by 9 and 5:

a) Divisible by 9: 30 (30 / 9 = 3 remainder 3), 405 (405 / 9 = 45), 972 (972 / 9 = 108), 865 (865 / 9 = 96 remainder 1)

b) Divisible by 5: 30 (30 / 5 = 6), 405 (405 / 5 = 81), 865 (865 / 5 = 173)

Therefore, the numbers that are divisible by 9 are 30, 405, 972, and 865, and the numbers that are divisible by 5 are 30, 405, and 865.

3) To factorize the number 660 into its prime factors, we can use the following steps:

a) Divide the number by the smallest prime number, which is 2. Repeat this step until the number is no longer divisible by 2. 660 / 2 = 330 330 / 2 = 165 165 / 2 = 82.5 (not divisible by 2)

b) Divide the number by the next prime number, which is 3. Repeat this step until the number is no longer divisible by 3. 165 / 3 = 55 55 / 3 = 18.33 (not divisible by 3)

c) Divide the number by the next prime number, which is 5. Repeat this step until the number is no longer divisible by 5. 55 / 5 = 11

d) The remaining number, 11, is a prime number.

Therefore, the prime factorization of 660 is 2 * 2 * 3 * 5 * 11.

4) To find the maximum number of identical sets that can be formed using 24 pencils and 32 pens, while using all the pencils and pens in each set, we need to find the greatest common divisor (GCD) of 24 and 32.

GCD(24, 32) = GCD(32, 24 % 32) = GCD(32, 24) = GCD(24, 32 % 24) = GCD(24, 8) = GCD(8, 24 % 8) = GCD(8, 0) = 8

Therefore, the maximum number of identical sets that can be formed is 8.

5) Let's solve the problem step by step:

Given that the number of apples is greater than 273 but less than 389, and the apples can be divided equally into 6, 7, or 8 piles.

a) Divisible by 6: The number should be divisible by both 2 and 3. The smallest number greater than 273 divisible by 6 is 276 (46 piles of 6 apples).

b) Divisible by 7: The number should be divisible by 7. The smallest number greater than 273 divisible by 7 is 280 (40 piles of 7 apples).

c) Divisible by 8: The number should be divisible by both 2 and 4. The smallest number greater than 273 divisible by 8 is 280 (35 piles of 8 apples).

Therefore, the woman brought 280 apples to the market.

Note: The calculations above assume that the number of apples is an integer.