What is the smallest positive integer that can be expressed as the sum of nine consecutive positive integers, the sum of ten consecutive positive integers, and the sum of eleven consecutive positive integers
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What is the smallest positive integer that can be expressed as the sum of nine consecutive positive integers, the sum of ten consecutive positive integers, and the sum of eleven consecutive positive integers
3. Consecutive Sums a. (4 pts) Write 90 as the sum of consecutive positive integers in as many wa...
3. Consecutive Sums a. (4 pts) Write 90 as the sum of consecutive positive integers in as many ways as possible. b. (4 pts) If a number can be written as n (d)(t) where d is an odd number of the form 2k + 1 and d is greater than 1, show symbolically how n can be written as the sum of consecutive numbers. Illustrate this with one example from part a. c. (4 pts) State a conjecture identifying the...
three consecutive even integers are such that the sum of the smallest and 3 times the second is 38 more than twice the third
three consecutive even integers are such that the sum of the smallest and 3 times the second is 38 more than twice the third. what are the integers?
find the sum of the reciprocals of two consecutive even integers if the smaller integer is x
find the sum of the reciprocals of two consecutive even integers if the smaller integer is x.
Prove by induction that the sum of any sequence of 3 positive consecutive integers is divisible...
Prove by induction that the sum of any sequence of 3 positive consecutive integers is divisible by 3. Hint, express a sequence of 3 integers as n+(n+1)+(n+2).
Eight consecutive three digit positive integers have the following property: each of them is divisible by...
Eight consecutive three digit positive integers have the following property: each of them is divisible by its last digit. What is the sum of the digits of the smallest of the eight integers? A 10 B 11 ะก 12 D 13 E 14
Challenge activity: A partition of a positive integer n is the expression of n as the sum of positive integers, whe...
Challenge activity: A partition of a positive integer n is the expression of n as the sum of positive integers, where order does not matter. For example, two partitions of 7 are 7 1+1+1+4 and 7=1+1+1+2+2. A partition of n is perfect if every integer from 1 to n can be written uniquely as the sum of elements in the partition. 1+1+1+4 is perfect since 1-7 are expressed only as 1, 1+1, 1+1+1, 4, 1+4, 1+1+4 and 1+1+1+4, but 1+1+1+2+2...
In how many ways can five positive integers be added such that their sum does not...
In how many ways can five positive integers be added such that their sum does not exceed ten?
Prove: If n and a are positive integers and n=(a^2+ 1)/2, then n is the sum...
Prove: If n and a are positive integers and n=(a^2+ 1)/2, then n is the sum of the squares of two consecutive integers (that is, n=k^2+ (k+1)^2 for some integer k).
Show that every positive integer n, there is a string of n consecutive integers where first...
Show that every positive integer n, there is a string of n consecutive integers where first integer is even, the second is divisible by a perfect square(other than 1), the third by a perfect cube(other than 1), etc..., and the nth is divisible by the nth power of an integer(other than 1). Then find an example for n = 3.
EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus,...
EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus, there are arbitrarily large gaps between primes. EXERCISE 1.12. Show that two integers are relatively prime if and only if there is no one prime that divides both of them.
3. Consecutive Sums a. (4 pts) Write 90 as the sum of consecutive positive integers in as many ways as possible. b. (4 pts) If a number can be written as n (d)(t) where d is an odd number of the form 2k + 1 and d is greater than 1, show symbolically how n can be written as the sum of consecutive numbers. Illustrate this with one example from part a. c. (4 pts) State a conjecture identifying the...
Eight consecutive three digit positive integers have the following property: each of them is divisible by its last digit. What is the sum of the digits of the smallest of the eight integers? A 10 B 11 ะก 12 D 13 E 14
Challenge activity: A partition of a positive integer n is the expression of n as the sum of positive integers, where order does not matter. For example, two partitions of 7 are 7 1+1+1+4 and 7=1+1+1+2+2. A partition of n is perfect if every integer from 1 to n can be written uniquely as the sum of elements in the partition. 1+1+1+4 is perfect since 1-7 are expressed only as 1, 1+1, 1+1+1, 4, 1+4, 1+1+4 and 1+1+1+4, but 1+1+1+2+2...
EXERCISE 1.28. Show that for every positive integer k, there exist k consecutive composite integers. Thus, there are arbitrarily large gaps between primes. EXERCISE 1.12. Show that two integers are relatively prime if and only if there is no one prime that divides both of them.
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