Question

7. At least how many numbers do you have to select from 1 to 1000 to guarantee that (a) there are two, such that their difference is a multiple of 6 . (b) there are two, such that their sum is a multiple of 6 .

(c) the sum of some (at least one) is a multiple of 6 (d) the product of all numbers is a multiple of 6 (e) two of the numbers add up to 111

Answer 1

To guarantee that two selected numbers meet the conditions stated, we would need at least 7 numbers, 2 numbers, 1 number, 3 numbers, and 2 numbers for conditions (a), (b), (c), (d) and (e) respectively.

Step-by-step explanation:

To guarantee that two selected numbers meet the conditions stated, we'll need to consider the properties of numbers. Let's solve each part:

(a) To ensure that their difference is a multiple of 6, we need at least 7 numbers. Take the first six numbers, 1 to 6, the differences are 1 to 5. By adding a seventh number, 7, we are guaranteed a difference of 6 with 1.

(b) To ensure that their sum is a multiple of 6, we need at least 2 numbers. One example is 2 and 4.

(c) In order to ensure that the sum of some numbers is a multiple of 6, we simply need 1 number, that being 6 itself.

(d) To ensure that the product of all numbers is a multiple of 6, we need three numbers: 2, 3, and any other number. 2*3 results in 6, which is a multiple of 6. Multiplying any other number will not affect it being a multiple of 6.

(e) To ensure two of the numbers add up to 111, we need two numbers. An example would be 56 and 55.

Learn more about Numbers properties