Question

Prove that there are infinitely many pairs $(m, n)$ such that $(m!)^n + (n!)^m + 1$ is divided by $n + m$.

score 158 · Answer 1 · 2023-10-03 03:56:45

158

Views

Prove that there are infinitely many pairs $(m, n)$ such that $(m!)^n + (n!)^m + 1$ is divided by $n + m$.

Published on 03 Oct 2023 - 3:56

score 90 · Answer 2 · 2023-10-04 20:34:33

90

Views

A number with digits $2$ and $5$ is divisible by $2^{2005}$

Published on 04 Oct 2023 - 20:34

score 36 · Answer 3 · 2026-04-02 14:06:51

36

Views

Intervals of length $k$ with at least $n$ distinct prime factors

Published on 02 Apr 2026 - 14:06

score 38 · Answer 4 · 2023-10-10 09:08:06

38

Views

Relation between two functions with divisibility properties

Published on 10 Oct 2023 - 9:08

score 12 · Answer 5 · 2026-02-27 10:43:36

12

Views

Proving $n$ divides $φ(a^n − 1)$ for any positive integer $a$ and $n$, where $a^n - 1 > 1$

Published on 27 Feb 2026 - 10:43

score 63 · Answer 6 · 2023-10-13 22:37:45

63

Views

(Dis)prove $\,a\mid b\,\Rightarrow\, a+b\mid b+c$

Published on 13 Oct 2023 - 22:37

score 88 · Answer 7 · 2026-04-01 22:18:05

88

Views

Proving divisibility of a sum $\sum_{k=0}^{\lfloor{n/2}\rfloor} (-1)^{k}\binom{n}{2k} 7^{k}$

Published on 01 Apr 2026 - 22:18

score 70 · Answer 8 · 2023-10-16 03:14:02

Pick a, b, c randomly without any conditions from the set ${1, 2, 3,..., 2019}$ What is the probability for $abc + bc + c$ to be divisible by 3?

score 68 · Answer 9 · 2026-04-02 05:05:20

68

Views

Criterion for a factorial ratio to be integral

Published on 02 Apr 2026 - 5:05

score 104 · Answer 10 · 2023-10-17 03:35:35

104

Views

Find all integers $n>1$ with the property that for each positive divisor $d$ of $n$, we also have that: $(d+2) \mid(n+2) $.

Published on 17 Oct 2023 - 3:35

score 71 · Answer 11 · 2023-10-17 10:26:02

71

Views

Show that the number isn't divisible by 341 (using congruences & Fermat's little theorem)

Published on 17 Oct 2023 - 10:26

score 123 · Answer 12 · 2023-10-18 16:05:11

123

Views

Prove that for all integers $a,b, b\neq 0,$ there exists $n$ with $v_2(n!) \equiv a\mod b$.

Published on 18 Oct 2023 - 16:05

score 36 · Answer 13 · 2026-04-02 07:06:30

36

Views

Prove that ${n\choose k}$ has at least k distinct prime divisors by proving an intermediate statement first

Published on 02 Apr 2026 - 7:06

score 85 · Answer 14 · 2023-10-22 03:21:46

85

Views

Prove that there exists a unique partition of $\mathbb{N}$ into A and B so that neither of $A\oplus A$ and $B\oplus B$ has a prime.

Published on 22 Oct 2023 - 3:21

score 102 · Answer 15 · 2023-10-22 07:16:13

102

Views

The exact meaning of $1/0$?

Published on 22 Oct 2023 - 7:16

List Question

Prove that there are infinitely many pairs $(m, n)$ such that $(m!)^n + (n!)^m + 1$ is divided by $n + m$.

A number with digits $2$ and $5$ is divisible by $2^{2005}$

Intervals of length $k$ with at least $n$ distinct prime factors

Relation between two functions with divisibility properties

Proving $n$ divides $φ(a^n − 1)$ for any positive integer $a$ and $n$, where $a^n - 1 > 1$

(Dis)prove $\,a\mid b\,\Rightarrow\, a+b\mid b+c$

Proving divisibility of a sum $\sum_{k=0}^{\lfloor{n/2}\rfloor} (-1)^{k}\binom{n}{2k} 7^{k}$

Pick a, b, c randomly without any conditions from the set ${1, 2, 3,..., 2019}$ What is the probability for $abc + bc + c$ to be divisible by 3?

Criterion for a factorial ratio to be integral

Find all integers $n>1$ with the property that for each positive divisor $d$ of $n$, we also have that: $(d+2) \mid(n+2) $.

Show that the number isn't divisible by 341 (using congruences & Fermat's little theorem)

Prove that for all integers $a,b, b\neq 0,$ there exists $n$ with $v_2(n!) \equiv a\mod b$.

Prove that ${n\choose k}$ has at least k distinct prime divisors by proving an intermediate statement first

Prove that there exists a unique partition of $\mathbb{N}$ into A and B so that neither of $A\oplus A$ and $B\oplus B$ has a prime.

The exact meaning of $1/0$?

Trending Questions

Popular # Hahtags

Popular Questions