54 Answered Questions for the topic Number Theory

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... more

08/12/19

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

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... more

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.

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?

07/05/19

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?

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... more

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

03/24/19

What is the multiplicative inverse of 3 module 46?

03/19/19

How many 2-digit positive integers are there?

How many 2-digit positive integers are there such that the product of their two digits is 24? The answer given is four. I'm not certain if I understand this question correctly and need some... more

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... more

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... more

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?

03/15/19

How many primes do I need to check to confirm that an integer $L$, is prime?

I recently saw the 1998 horror movie "Cube", in which a character claims it is humanly impossible to determine, by hand without a computer, if large (in the movie 3-digit) integers are prime... more

03/14/19

How many 2-digit positive integers are there?

How many 2-digit positive integers are there such that the product of their two digits is 24? The answer given is four. I'm not certain if I understand this question correctly and need some... more

03/12/19

Find a prime factor of $7999973$ without a calculator?

How would you go about finding prime factors of a number like $7999973$? I have trivial knowledge about divisor-searching algorithms.

01/18/19

What are the last digits of 3^3^3^3

If Xn=3^3^3.(n times)..^3 with n bigger than 10, show that their 10 last digits are the same. I think we should use Euler Fermat's theorem, and calculate with modulo(10^10) and somehow prove it by... more

10/26/18

Find the value of an integer a such that a^2 +6a +1 is a perfect square.

I was able to solve this but it required me using hit and trial at one step. I was wondering if i could find a more solid method to solve it.p.s. this is the first time im asking a question here so... more

03/30/18

what is the prime factorization for 27

 i dont know

11/06/17

Why is 2 a controversial number?

I have heard people say 2 is not a prime number and some go as far to say that 2 is not a number all together. What is the meaning behind this? Have I been lied to my entire life?

10/08/17

Explain why gcd(a, b) = 1.

If 1 = sa + tb, where a, b, st ∈ Z, explain why gcd(a, b) = 1.    
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