2
votes
3answers
2k views

Find the sum of all the integers between 1 and 1000 which are divisible by 7

How can I work this one out (with workings)? "Find the sum of all the integers between 1 and 1000 which are divisible by 7" Thanks!
4
votes
1answer
134 views

Solve $a^3 + b^3 + c^3 = 6abc$

Find solutions for $a^3 + b^3 + c^3 = 6abc$ in $\mathbb{N}$, such that $gcd(a,b,c) = 1$, except for $(1,2,3)$ and its permutations. Using trial and error I found out that if $a,b,c$ are solution ...
1
vote
1answer
109 views

Techniques to prove properties of a sequence

What techniques/methods can be used to prove that the sequence produced by $n\cdot (n+1)\cdot (2\cdot n+1)/6$ contains only one square ($4900$) greater than 1? While this particular sequence is an ...
3
votes
6answers
201 views

Show that $121^{n}-25^{n}+1900^{n}-(-4)^{n}$ is divisible by 2000.

My question comes from the British Mathematical Olympiad (Round 1) paper from 2000: Show that $121^{n}-25^{n}+1900^{n}-(-4)^{n}$ is divisible by 2000 for any natural $n$. My immediate idea was to ...
0
votes
1answer
256 views

How can I find all increasing sequences $\{a_i\}_{i=1}^{\infty}$ such that $d(x_1+x_2+\cdots+x_k)=d(a_{x_{1}}+a_{x_{2}}+\cdots + a_{x_{k}})$?

How can one find all increasing sequences $\{a_i\}_{i=1}^{\infty}$ such that $$d(x_1+x_2+\cdots+x_k)=d(a_{x_{1}}+a_{x_{2}}+\cdots + a_{x_{k}}),$$ holds for all $k$-tuples $(x_1,x_2,\cdots,x_k)$ of ...