Question

$A$ has $2$ coins and $B$ has $n$ coins. $B$ wants to buy some pencils, which come in two different packages. One package of pencils costs $6$ coins for $7$ pencils, and the other package of pencils costs $10$ coins for $12$ pencils. $B$ notes that he can spend all $n$ of his coins on some combination of pencil packages to get $p$ pencils. However, if $A$ gives his two coins to $B,$ he then notes that he can instead spend $n+2$ of his coins on some combination of pencil packages to get fewer than $p$ pencils. The smallest value of $n$ for which this is possible, is equal to

• A. $100$
• B. $120$
• C. $60$
• D. $45$

Answer 1

The correct option is A $100$

Let $B$ buy $a$ packages of $7$ and $b$ packages of $12$ in first case,
$c$ packages of $7$ and $d$ packages of $12$ in second case.

$6a+10b=n,$
$6c+10d=n+2$
and $7a+12b>7c+12d$
$\Rightarrow 3(c-a)-5(b-d)=1$
and $12(b-d)>7(c-a)$ & $b-d>0$
$(c-a,b-d)=(2,1),(7,4),(12,7),(17,10)$
$(17,10)$ is the first pair for $12(b-d)>7(c-a)$
Hence, $b\ge 10$ and $n\ge 100$
Smallest $n=100$