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.
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Show that every positive integer n, there is a string of n
consecutive integers where first...
Find all three consecutive positive integers such that the first is divisible by 3, the second...
Find all three consecutive positive integers such that the first is divisible by 3, the second is divisible by 5 and the third is divisible by 53.
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...
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.
please answer questions #7-13 7. Use a direct proof to show every odd integer is the...
please answer questions #7-13
7. Use a direct proof to show every odd integer is the difference of two squares. [Hint: Find the difference of squares ofk+1 and k where k is a positive integer. Prove or disprove that the products of two irrational numbers is irrational. Use proof by contraposition to show that ifx ty 22 where x and y are real numbers then x 21ory 21 8. 9. 10. Prove that if n is an integer and 3n...
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...
For Exercises 1-15, prove or disprove the given statement. 1. The product of any three consecutive...
For Exercises 1-15, prove or disprove the given
statement.
1. The product of any three consecutive integers is even.
2. The sum of any three consecutive integers is
even.
3. The product of an integer and its square is
even.
4. The sum of an integer and its cube is even.
5. Any positive integer can be written as the sum of
the squares of two integers.
6. For a positive integer
7. For every prime number n, n +...
Tems.] Use the second principle of induction to prove that every positive integer n has a...
Tems.] Use the second principle of induction to prove that every positive integer n has a factorization of the form 2m, where m is odd. (Hint: For n > 1, n is either odd or is divisible by 2.)
According to the Journal of Irreproducible Results, any obtuse angle is a right angle
According to the Journal of Irreproducible Results, any obtuse angle is a right angle! Here istheir argument.Given the obtuse angle x, we make a quadrilateral ABCD with DAB = x, and ABC =90◦, andAD = BC. Say the perpendicular bisector toDC meets the perpendicular bisector toAB at P. ThenPA = PB andPC = PD. So the trianglesPADandPBC have equal sidesand are congruent. Thus PAD = PBC. But PAB is isosceles, hence PAB = PBA.Subtracting,...
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On...
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose...
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose r is a real number other than 1. Prove using mathematical induction that for every nonnegative integer n, = 1-r^n+1/1-r. 3) Prove using mathematical induction that for every nonnegative integer n, 1 + i+i! = (n+1)!. 4) Prove using mathematical induction that for every integer n>4, n!>2^n. 5) Prove using mathematical induction that for every positive integer n, 7 + 5 + 3 +.......
Find all three consecutive positive integers such that the first is divisible by 3, the second is divisible by 5 and the third is divisible by 53.
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...
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.
please answer questions #7-13
7. Use a direct proof to show every odd integer is the difference of two squares. [Hint: Find the difference of squares ofk+1 and k where k is a positive integer. Prove or disprove that the products of two irrational numbers is irrational. Use proof by contraposition to show that ifx ty 22 where x and y are real numbers then x 21ory 21 8. 9. 10. Prove that if n is an integer and 3n...
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...
For Exercises 1-15, prove or disprove the given
statement.
1. The product of any three consecutive integers is even.
2. The sum of any three consecutive integers is
even.
3. The product of an integer and its square is
even.
4. The sum of an integer and its cube is even.
5. Any positive integer can be written as the sum of
the squares of two integers.
6. For a positive integer
7. For every prime number n, n +...
Tems.] Use the second principle of induction to prove that every positive integer n has a factorization of the form 2m, where m is odd. (Hint: For n > 1, n is either odd or is divisible by 2.)
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
Show that if n is a positive integer and a and b are integers relatively prime to 1 such that (On(a), On(b))1, then
Prove using mathematical induction that for every positive integer n, = 1/i(i+1) = n/n+1. 2) Suppose r is a real number other than 1. Prove using mathematical induction that for every nonnegative integer n, = 1-r^n+1/1-r. 3) Prove using mathematical induction that for every nonnegative integer n, 1 + i+i! = (n+1)!. 4) Prove using mathematical induction that for every integer n>4, n!>2^n. 5) Prove using mathematical induction that for every positive integer n, 7 + 5 + 3 +.......
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