In this question we will write $( x , y )$ for a vector instead of the usual $\binom { x } { y }$ notation. So, for example, $3 ( 2,1 ) = ( 6,3 )$.
This question is about vectors $( x , y )$ where $x$ and $y$ are whole numbers, and whether or not such vectors can be written in the form $a ( 5,0 ) + b ( 0,7 ) + c ( 2,1 )$ where $a$ and $b$ and $c$ are whole numbers each greater than or equal to zero.
We will consider the set $S$ of vectors $( p , q )$ with $0 \leq p \leq 4$ and $0 \leq q \leq 6$, with $p$ and $q$ whole numbers.
Then, by considering the vectors $( x , y ) , ( x , y ) - ( 2,1 ) , ( x , y ) - 2 ( 2,1 ) , \ldots , ( x , y ) - 34 ( 2,1 )$, we will find conditions on $x$ and $y$ that imply that $( x , y )$ can be written in the form $a ( 5,0 ) + b ( 0,7 ) + c ( 2,1 )$ where $a$ and $b$ and $c$ are whole numbers each greater than or equal to zero.
(i) Consider the set $S$ of vectors $( p , q )$ with $0 \leq p \leq 4$ and $0 \leq q \leq 6$, with $p$ and $q$ whole numbers.
(a) How many vectors are in $S$ ?
(b) Explain why for any vector $( x , y )$ with $x$ and $y$ whole numbers, we can find whole numbers $a$ and $b$, and a vector $\mathbf { v }$ in $S$ such that
$$( x , y ) = a ( 5,0 ) + b ( 0,7 ) + \mathbf { v }$$
In the rest of this question, we'll call such a vector $\mathbf { v }$ the residue of ( $x , y$ ), and we will assume that the residue is uniquely defined for each vector $( x , y )$.
(ii) Consider vectors $( x , y ) + k ( 2,1 )$ and $( x , y ) + m ( 2,1 )$ where $k$ and $m$ are whole numbers.
(a) Prove that if these vectors have the same residue (as defined in the previous part of the question), then $( k - m )$ is a multiple of 35 .
(b) Explain why the vectors $( x , y ) , ( x , y ) - ( 2,1 ) , ( x , y ) - 2 ( 2,1 ) , \ldots , ( x , y ) - 34 ( 2,1 )$ all have different residues.
(iii) Hence show that if $x$ is at least 68 and $y$ is at least 34 , with $x$ and $y$ whole numbers, then the vector $( x , y )$ can be written in the form $a ( 5,0 ) + b ( 0,7 ) + c ( 2,1 )$ where $a$ and $b$ and $c$ are whole numbers each greater than or equal to zero.
You do not need to find the values of $a$ and $b$ and $c$.
(iv) A student claims: "if $x$ and $y$ are whole numbers with $x > 0$ and $y > 0$ and $x + y$ at least 102 then $( x , y )$ can be written in the form $a ( 5,0 ) + b ( 0,7 ) + c ( 2,1 )$ where $a$ and $b$ and $c$ are whole numbers each greater than or equal to zero."
Is the student's claim correct? Justify your answer.