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Questions tagged [elementary-number-theory]

For questions on introductory topics in number theory, such as divisibility, prime numbers, gcd and lcm, congruences, linear Diophantine equations, Fermat's and Wilson's theorems, the Chinese Remainder theorem, primitive roots, quadratic congruences, quadratic number fields, Pell's equations, and related topics.

For questions on introductory topics in number theory, such as divisibility, prime numbers, gcd and lcm, congruences, linear Diophantine equations, Fermat's and Wilson's theorems, the Chinese Remainder theorem, primitive roots, quadratic congruences, quadratic number fields, Pell's equations, and related topics.

For more advanced topics, use relevant tags, e.g.

37415 questions
57
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8 answers

Is there a formula to calculate the sum of all proper divisors of a number?

I don't need to list all the proper divisors; I just want to compute their sum. While checking and summing up all proper divisors isn't an issue for small numbers, it becomes significantly slower for larger numbers. Any suggestions? Thanks!
roxrook
  • 12,081
52
votes
3 answers

How to weigh up to 100kg with 5 weights

1) You are a shopkeeper who is selling sugar between 1-100 kg .Now you have to design 5 weights in such a way that any integer weight between 1-100 can be measured in a single attempt ,without using more than 5 weights.You can't repeat weights. He…
mrigendra
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13 answers

How can I write the numbers 5 and 7 as some sequence of operations on three 9s?

I want to make the numbers $1, 2, ..., 9$ using exactly three copies of the number $9$ and the following actions: addition, subtraction, multiplication, division, squaring, taking square roots, and other action. How can we make the numbers $5$ or…
elham
  • 1,195
47
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6 answers

Second part of the factorial sum divisibility question

Which primes $p$ divide the sum of factorials $1! + 2! + 3! + 4! + 5! + \cdots + (p-1)!$? This is related to my previous question.
user58512
47
votes
2 answers

Proof: How many digits does a number have? $\lfloor \log_{10} n \rfloor +1$

I read recently that you can find the number of digits in a number through the formula $\lfloor \log_{10} n \rfloor +1$ What's the logic behind this rather what's the proof?
Jeel Shah
  • 9,306
40
votes
3 answers

Why is $(2+\sqrt{3})^{50}$ so close to an integer?

I just worked out $(2+\sqrt{3})^{50}$ on my computer and got the answer $39571031999226139563162735373.999999999999999999999999999974728\cdots$ Why is this so close to an integer?
user1510374
  • 902
38
votes
8 answers

Is the number 100k+11 ever a square?

Is the number 100k+11 ever a square? Obviously all such numbers end in 11. Can a square end with an 11?
Adam
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36
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7 answers

Why is the last digit of $n^5$ equal to the last digit of $n$?

I was wondering why the last digit of $n^5$ is that of $n$? What's the proof and logic behind the statement? I have no idea where to start. Can someone please provide a simple proof or some general ideas about how I can figure out the proof myself?…
Jeel Shah
  • 9,306
36
votes
6 answers

How can we measure how "irrational" a number is?

I wonder, can we, for instance, say that $\pi$ is more irrational than $e$? Or that $e$ is more irrational than $\sqrt{2}$? What kind of irrationality measurement can we use to say that $A$ is (much) more irrational than $B$? (in particularly…
Gary B
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35
votes
1 answer

If $2^{2^j} a + 1$ divides $c^{2^j}+1$ for fixed $a,c$ and all $j$, then $a=1$,$c=2^l$ for some odd $l$, or $a=0$.

I have a very hard problem: Prove that, if $2^{2^j} a + 1$ divides $c^{2^j}+1$ for fixed integers $a,c$ and all nonnegative integers $j$, then $a=1$ and $c=2^l$ for some odd positive integer $l$, or else $a=0$. Here is my progress on the problem…
Is Ne
  • 2,668
33
votes
9 answers

Prove that $\gcd(M, N)\cdot \mbox{lcm}(M, N) = M \cdot N$.

I'm not sure how to go about this proof. I just need help getting started. Is there a way to prove it algebraically?
Susan
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33
votes
8 answers

Write 100 as the sum of two positive integers

Write $100$ as the sum of two positive integers, one of them being a multiple of $7$, while the other is a multiple of $11$. Since $100$ is not a big number, I followed the straightforward reasoning of writing all multiples up to $100$ of either…
Lanner
  • 333
32
votes
4 answers

31,331,3331, 33331,333331,3333331,33333331 are prime

31,331,3331, 33331,333331,3333331,33333331 are prime. This law can continue it? Will there emerge a composite number? Without using a computer how to judge.
tianzhidaosunyouyu
  • 1,786
32
votes
7 answers

Why is every answer of $5^k - 2^k$ divisible by 3?

We have the formula $$5^k - 2^k$$ I have noticed that every answer you get from this formula is divisible by 3. At least, I think so. Why is this? Does it have to do with $5-2=3$?
iDivide
  • 351
32
votes
5 answers

What's the smallest number with first digit 1 that triples when this digit is moved to the end?

There's this homework question I have, and while I know people generally don't like these, I would like a hint on how to get started please. A positive integer begins with the digit 1 when written in decimal. When this digit is transferred to the…
Kbot
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