Can you check if my proof is right?
Theorem. $\forall x\geq8, x$ can be represented by $5a + 3b$ where $a,b \in \mathbb{N}$.
Base case(s): $x=8 = 3\cdot1 + 5\cdot1 \quad \checkmark\\
x=9 = 3\cdot3 + 5\cdot0 \quad \checkmark\\
x=10 = 3\cdot0 + 5\cdot2 \quad \checkmark$
Inductive step:
$n \in \mathbb{N}\\a_1 = 8, a_n = a_1 + (x-1)\cdot3\\
b_1 = 9, b_n = b_1 + (x-1)\cdot3 = a_1 +1 + (x-1) \cdot 3\\
c_1 = 10, c_n = c_1 + (x-1)\cdot3 = b_1 + 1 + (x-1) \cdot 3 = a_1 + 2 + (x-1) \cdot 3\\
\\
S = \{x\in\mathbb{N}: x \in a_{x} \lor x \in b_{x} \lor x \in c_{x}\}$
Basis stays true, because $8,9,10 \in S$
Lets assume that $x \in S$. That means $x \in a_{n} \lor x \in b_{n} \lor x \in c_{n}$.
If $x \in a_n$ then $x+1 \in b_x$,
If $x \in b_x$ then $x+1 \in c_x$,
If $x \in c_x$ then $x+1 \in a_x$.
I can’t prove that but it’s obvious. What do you think about this?
Proof by induction.
For the base case $n=8$ we have $8=5+3$.
Suppose that the statement holds for $k$ where $k\gt 8$. We show that it holds for $k+1$.
There are two cases.
1) $k$ has a $5$ as a summand in its representation.
2) $k$ has no $5$ as a summand in its representation.
For case 1, we delete “that $5$” in the sum representation of $k$ and replace it by two “$3$”s ! This proves the statement for $k+1$.
For case 2, since $k\gt 8$, then $k$ has at least three “$3$”s in its sum representation. We remove these three $3$’s and replace them by two fives! We obtain a sum representation for $k+1$. This completes the proof.
Write $x=8n+k$ for $0\leq k<8$ and $n\geq 1$. Because $8n=(3+5)n$, the problem is clear if $k=0,3,5$. Let’s consider the remaining 5 cases:
\begin{align*}
k=1 & \implies 8n+1=5(n-1)+3(n+2),\\
k=2 & \implies 8n+2=5(n+1)+3(n-1),\\
k=4 & \implies 8n+4=5(n-1)+3(n+3),\\
k=6 & \implies 8n+6=5n+3(n+2),\\
k=7 & \implies 8n+7=5(n+2)+3(n-1).
\end{align*}
Note that $k=4$ is based on $k=1$ (just add one more $3$) and, similarly, $k=6$ and $k=7$ follow from $k=1$ and $k=2$, respectively. So in fact, we have really only considered 2 cases.
You are almost done, actually you can prove it even without induction:
If $\forall x \ge 8$ and you must prove that $x \in \mathbb{N} \land x = 5a + 3b$
Let $x_1$ be 8 and consider your base cases:
$x_1 = 8 = 3 ⋅ 1 + 5 ⋅ 1$
$x_2 = 9 = 3⋅3 + 5 ⋅ 0$
$x_3 = 10 = 3⋅0 +5⋅2$
This is true for the first 3 $n$ (1, 2 and 3) of $x_n$.
Now, consider the following:
$x_4 = 11 = x_1 + 3$
$x_5 = 12 = x_2 + 3$
$x_6 = 13 = x_3 + 3$
$x_7 = 14 = x_4 + 3$
$x_8 = 15 = x_5 + 3$
$…$
$x_n = x_{n-3} + 3 \,\,\,\,\,\,\forall n \gt 3$
As you can see from the pattern, it’s obvious that you are summing only multiples of 3 basing on your previous results (which were also sums of multiples of threes and multiples of fives).
And you may also note that as soon as you sum 5 threes ($3+3+3+3+3 = 15 = 3⋅5$) you can safely replace them with 3 fives ($5 + 5 + 5 = 15 = 3⋅5$)
The previous equation leads to a recurrence relation and you can say with certainty that your statement is true $\forall x \ge 8$
I would avoid induction and use the elementary Euclidean division algorithm (Eda).
Let $n\geq8$ be an integer. Then, by the Eda, there exist integer $q$ and $r$ such that $r\in\{0,1,2\}$ and $$n=3q+r.$$
If $r=0$, we are done since $n=3q$.
If $r=1$, then $q\geq3$ (because $n\geq8$). Hence $n=3(q-3)+10$.
If $r=2$, then $q\geq2$ (because $n\geq8$). Hence $n=3(q-1)+5$.
This is some mixture of Bezout’s Lemma and Frobenius coin problem. Frobenius coin problem claims that if $a_1$ and $a_2$ are coprime numbers then every number bigger or equal to $(a_1-1)(a_2-1)$ can be written in the form $a_1x + a_2y$ for some non-negative integers $x,y$.
Write $a_1=3$ and $a_2=5$ in you case and you’ve got the answer. Anyway a inductive proof wouldn’t be much harder.
1.if the integer can be written as $3s$, then set $a=0$
2.if the integer can be written as $3s+1$, set $a=2$, $10+3b=3s+1$,b=s-3
3.if the integer can be written as $3s+2$, set $a=1$, $b=s-1$
I am sure you have got the idea.
Your base case work is fine.
For the induction, you can assume (1) $k\ge 10$ and (2) all cases from $n=8$ to $n=k$ hold.
Then it is simply a matter of noting that we have $k-2=5a+3b$ and so we can say that $k+1 = 5a+3(b+1)$ as required.
Any number is of the form $5k+1$, $5k+2$, $5k+3$, $5k+4$ and $5k+5$.
If $n=5k+3$ then nothing to prove.
$n=5k+1 \implies n=5(k-1)+2\cdot3$
$n=5k+2 \implies n=5(k-2)+4\cdot3$
$n=5k+4\implies n=5(k-4)+8\cdot3$
$n=5k+5\implies n=5(k-2)+5\cdot 3$
Hence for all $k\ge5$ the result holds.
Hence for all $n\ge 25$ the result hold.
What is left is to check the remaining cases and that is also true.
Hence the proposition is true.
Let $P(m)$ be the proposition that $k$ can be represented by a sum of fives and threes for all $m-2 \le k \le m$. $P(10)$ is true, since $8=5+3$, $9=3+3+3$, and $10=5+5$. This is the base case. And if $P(m)$ is true, then so is $P(m+1)$, since (applying the inductive hypothesis at the *): $$m+1=(m-2)+3 =^*(3a+5b)+3=3(a+1)+5b.$$ By induction on $m$, then, $P(m)$ is true for all $m\ge 10$; hence $k$ can be represented by a sum of fives and threes for all $k \ge 8$.
Really late to the party, but here is a very short and sweet approach:
Claim: For each integer $n\geq 8$, let $S(n)$ be the statement that $n$ is expressible as a sum of $3$’s and $5$’s.
Base step: Since $8=3+5, 9=3+3+3, 10=5+5, S(8), S(9)$, and $S(10)$ are true.
Inductive step ($S(k)\to S(k+3)$): Let $k\geq 8$ and assume that $S(k)$ is true, where for some non-negative integers, $a,b,k = a\cdot 3+b\cdot 5$. Then $$
k+3=(a+1)\cdot 3+b\cdot 5,
$$
a sum of $3$’s and $5$’s, so $S(k+3)$ is true.
By mathematical induction, three cases of induction actually, for all $n\geq 8, S(n)$ is true. $\blacksquare$
Lets call $T$ every multiple of three: $T$ satisfies the condition. Numbers are $T, T-1,$ or $T-2$.
for $n > 3\cdot 2+1$:
$$n=T \implies n=3k $$
$$n=T-1 \implies n= 3\cdot 2 – 1 + 3(k-2) = 5 + 3(k-2)$$
$$n=T-2 \implies n= 3\cdot 4 – 2 + 3(k-4) = 10 + 3(k-2)$$
I’m attempting to answer the question intuitively:
Every third number is a multiple of 3, and can definitely be written as a sum of 3s and 5s:
$3x$ + $5$ x $0$.
Now we need to deal with numbers $y$ for which $y mod 3 = 1$ and $y mod 3 = 2$
For $y mod 3 = 2$:
We know that $5 mod 3 = 2$, so 5 x 1 + 3$x$ should give us all numbers $y$ where $y mod 3 =2$
For $y mod 3 = 1$:
We know that $5$ x $2 mod 3 = 1$, so 5 x 2 + 3$x$ should give us all numbers $y$ where $y mod 3 =1$
The smallest case in which we can use this proof is 5 x 2 + 3 x 0 = 10
We can show it is possible for 8 and 9 as 5+3 =8 and 3 x 3 =9
Hence, we can show that is possible for all numbers greater than or equal to 8.
This is also proves that the maximum multiplier we would need for 5 is 2.