64 Answered Questions for the topic Number Theory

12/29/20

#### Proof problem about gcd

Prove the following statements:
gcd(a,b) = gcd(a-b,b)
if gcd(a,b)=d, then exists such x, y ∈ Z, that ax+by=d

12/29/20

#### Proof problem about divisors

Prove that number n = p1z1 p2z2 ... pkzk has (z1+1)(z2+1)...(zn+1) divisors.

12/29/20

#### Mathematical number theory proof problem

If number n is divisible by a and n = p1z1 p2z2 ... pkzk, prove that n = p1l1 p2l2 ... pklk with li ≤ zi with all i = 1,...,k

Number Theory Math

12/21/20

#### How many pairs of integers (m,n) satisfy the equation m + n = mn?

How many pairs of integers (m,n) satisfy the equation m + n = mn?

10/27/20

#### Use congruences to prove that the divisibility test 5 works.

Divisibility rule for 5: Let a be any integer, 5 divides a if and only if the last digit of a is 0 or 5.

Number Theory Math

10/14/20

#### Is it always possible to find a positive integer m such that a is congruent to b mod m?

For any two integers a and b, is it always possible to find a positive integer m such that a ≡b mod m? Either show that it is possible or find a counterexample to show that it is not.

09/03/20

#### Find the greatest common divisor of the numbers 2^(120) − 1 and 2^(100) − 1

Find the greatest common divisor of the numbers 2120 − 1 and 2100 − 1

Number Theory Algebra 2

08/04/20

#### summation of multiples of odd natural numbers

what is the sum of all two odd natural number multiples in 'n' natural numbers, where n>=9.

Number Theory Infinite Disjoint Arithmetic Progressions

03/25/20

#### Disjoint Covering systems

Can their exist a disjoint covering system of Z+, made up of an infinite quantity of equivalence classes?

#### Proof that pairs of numbers whose GCF=1 can generate all subsequent integers

Prove that any linear combination of pairs of relatively prime numbers (that is pairs of numbers whose GCF is 1) will, beyond a certain number, generate all subsequent integers.*This does not hold...
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Number Theory

11/18/19

#### Math number theory

Three track team members are running on the school track. It takes Jess 4 minutes to complete a lap, jen 5 minutes, and Jill 6 minutes. If they start at the same point at the same time, how long...
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Number Theory Algebra 2

08/12/19

#### Prove that Z [i] is a commutative ring with unity.

Number Theory Algebra

07/08/19

#### What is the condition for the equation: ca^2 - (c+1)ab + b^2 to be non-negative? Is this generalizable for anything other than the trivial case of c=1?

When c = 1, the equation reduces to a^2 - 2ab + b^2 which is (a-b)^2 which is non-negative. What happens for other positive values of c (c can never be negative in my equation), assuming a and b...
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07/07/19

#### The units digit of a perfect square x is 4. What is the parity of the tens digit of x?

This is from my Math Challenge class. I have no idea of how to solve this. Please help.

Number Theory Math Competition Prep

07/06/19

#### Find the least natural number such that the number is divisible by 75 and all its digits are 1 or 0 only.

Is there any trick to be able to do this fast?

07/05/19

#### How do I solve this problem regarding parity from my Math Challenge class?

The remainder when the product 1492 x 1776 x 1812 x 1996 is divided by 5 is A. 0B. 1 C. 2 D. 3 E. 4*what's the shortcut to figuring out the answer for problems like this?*

07/05/19

#### How do I solve this problem regarding divisibility from my Math Challenge class?

Which of the following numbers is divisible by 2 and 3, but not divisible by 4?
3346
2376
5554
3282
9996

07/05/19

#### How do I solve this problem regarding divisibility from my Math Challenge class?

What remainder does the number 123456789 leave after division by 8? Is there a trick to solving this?

#### How do I solve this problem regarding divisibility from my Math Challenge class?

What remainder does the number (-50) leave after division by 7? The answer is 6. Why?

Number Theory Algebra 2

05/05/19

#### Show that if x,y,z are integers such that x^3 + 5y^3 = 25z^3, then x = y = z = 0.

I tried using mod 5 to get that x is divisible by 5. I don't know much about infinite descent, so I do not think that is the way to go.The hint says "If I give you a non-trivial solution, can you...
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04/29/19

#### a,b,c are positive real numbers with abc=1 then prove that (a / a+b⁴+c⁴) + (b / a⁴+b+c⁴) + (c / a⁴+b⁴+c)＜=1

a,b,c are positive real numbersabc=1prove that (a / a+b⁴+c⁴) + (b / a⁴+b+c⁴) + (c / a⁴+b⁴+c)＜=1

Number Theory

03/24/19

#### What is the multiplicative inverse of 3 module 46?

03/19/19

#### Numbers that are the sum of the squares of their prime factors?

A number which is equal to the sum of the squares of its prime factors with multiplicity:
- $16=2^2+2^2+2^2+2^2$
- $27=3^2+3^2+3^2$
Are these the only two such numbers to exist?
There has to be...
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03/19/19

#### Can it be proven/disproven that there are highly composite numbers that prime-factorize into larger primes such as $9999991$?

Of course, following the rules found by Ramanujan, such a highly composite number would need to factorize into all primes ascending up to 9999991 (with descending powers as the primes progress) so...
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03/18/19

#### When is $4n^4+1$ prime?

Find all natural numbers $n$ such that $4n^4+1$ is prime.
$4n^4+1$ is obviously prime when $n=1$. But can we prove that no other $n$ works?

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