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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

User Psulek
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Final answer:

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

User Mibbit
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