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How do you prove that among any set of 51 different positive integers whose values are at most 100 there is always a pair of numbers that differ by exactly 50?
To prove this statement, we can consider the pigeonhole principle. If we divide the set of 51 integers into two groups based on their remainders when divided by 50 (i.e., group A for remainders 0-49 and group B for remainders 50-99), there are only 50 possible remainders. Therefore, by the pigeonhole principle, at least one of the groups must contain at least two integers. These two integers must differ by exactly 50 since they belong to different remainder groups.
What else can I help you with?
Are integers counting numbers?
No- not exactly. Negative integers are not counting numbers. Positive integers are identified with counting numbers. Many authors like to start with zero as a counting number.
How many positive integers with exactly three decimal digit positive integers between 100 and 999 inclusive divisible by 7?
Of the 729 numbers that satisfy the requirement of positive integers, 104 are divisible by 7.
How do signed numbers differ from integers?
Positive signed numbers with have a + Positive integers will not.
Are negative numbers even integers?
A negative number can be even but doesn't have to be, and it can be an integer but doesn't have to be. Exactly like positive numbers.
How are integers and natural numbers different?
Integers are all positive and negative whole numbers, and natural numbers are all positve whole numbers including zero. So, natural numbers is a subset of integers.
Are integers counting numbers?
No- not exactly. Negative integers are not counting numbers. Positive integers are identified with counting numbers. Many authors like to start with zero as a counting number.
How many positive integers with exactly three decimal digit positive integers between 100 and 999 inclusive divisible by 7?
Of the 729 numbers that satisfy the requirement of positive integers, 104 are divisible by 7.
How do signed numbers differ from integers?
Positive signed numbers with have a + Positive integers will not.
How do you compare integers with positive numbers?
Some integers are positive numbers.Some integers are not positive numbers.Some positive numbers are integers.Some positive numbers are not integers.They are two sets whose intersection is the set of counting numbers.
Is positive integers belong to natural numbers?
Positive integers are (not is!) a proper subset of natural numbers. The natural numbers comprise positive integers and zero.
How are integers and natural numbers different?
Integers are all positive and negative whole numbers, and natural numbers are all positve whole numbers including zero. So, natural numbers is a subset of integers.
Are negative numbers even integers?
A negative number can be even but doesn't have to be, and it can be an integer but doesn't have to be. Exactly like positive numbers.
Are positive and negative numbers integers?
yes integers are all numbers negative and positive
What are distinct integers?
Integers are whole numbers, positive, negative or zero. Distinct merely means different.
When two different prime numbers are multiplied can the product be a negative integer?
Since prime numbers are defined as positive integers, any product of prime numbers will be positive.
What are distinct positive integers?
The set of positive integers is {1,2,3,4,5,...}. When referring to numbers, distinct simply means different from each other e.g. 2,6,7 and 9 are distinct positive integers but 2,6,6 and 9 are not distinct since two of them are equal.
Are natural numbers counting numbers and positive numbers different names for the same set of numbers?
Natural numbers are the same as counting numbers, but the term positive numbers means something else. Natural or counting numbers are positive integers, but the category of positive numbers includes both integers and fractions, as long as they are greater than zero.
