# Tagged Questions

Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

2answers
74 views

### If $n>1$ is an integer and $0<x\le n$ and $0<y\le n$, Prove that the equation $x^n+y^n=z^n$ has no solution.

If $n>1$ is an integer and $0<x\le n$ and $0<y\le n$, Prove that the equation $x^n+y^n=z^n$ has no solution. My work: This is obvious for integers by Fermat's Last Theorem. I also think that ...
1answer
59 views

### For given positive integers $n,k$ prove that there always exists some positive integer $x$ for which $2^n\mid \dfrac{x(x+1)}{2}-k$

For given positive integers $n,k$ prove that there always exists some $x$ for which $2^n \mid \dfrac{x(x+1)}{2}-k.$ My work: $\dfrac{x(x+1)}{2}$ is the sum of all positive integers upto $x$. Now, ...
2answers
418 views

6answers
2k views

### Prove that a number is even, given the cube is even

If it is known that $x^3$ is even, can we say that $x$ is even? It seems to be the case because an odd*odd*odd=odd (if we are dealing with natural numbers). But is there a proof?
2answers
295 views

### For a positive integer $n$ both $5n+1$ and $7n+1$ are perfect squares. Show that $n$ is divisible by 24.

My try: $5n + 1 = k^2$ $7n +1 = \frac{7k^2-2}5$ Just don't know how to proceed after this. Please help.
2answers
268 views

### The number of ordered triples $(a, b, c)$ of positive integers which satisfy the simultaneous equations $ab + bc = 44$, $ac + bc = 33$

My try: Subtracting the eqns: $a(b-c) = 11$ $a=1,b-c=11$ OR $a=11,b-c=1$ Substituting these values back int the original eqn. does not give an integral answer. Thus number of ordered ...
1answer
90 views

### Prove or disprove the following statement. $7 \ | \ (x^3 + x^2 + x + 2)$, where $x$ is an odd integer

We're learning about modulus and division (Discrete mathematics and proofs course). I'm not exactly sure how to tackle this sort of problem, is there some sort of property of cubic functions ...
2answers
47 views

### For each of the following values of ($a,b$), find the largest number that is not of the form $ax+by$ with $x\geq 0$ and $y \geq 0$.

For each of the following values of ($a,b$), find the largest number that is not of the form $ax+by$ with $x\geq 0$ and $y \geq 0$. $(i) (a,b) = (3,7)$ $(ii) (a,b) = (5,7)$ $(iii) (a,b) = (4,11)$ ...
2answers
177 views

### Number-Theoretic Coin Puzzle

There are three piles of coins. You are allowed to move coins from one pile to another, but only if the number of coins in the destination pile is doubled. For example, if the first pile has 6 coins ...