54 Answered Questions for the topic Number Theory

24d

#### 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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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.

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...
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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

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...
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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?

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...
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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...
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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...
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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...
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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?

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