Jax bought exactly enough trees to plant eight equal rows. Then one tree died and couldn't be planted, but he still had enough trees left to plant exactly nine equal rows. After that, a tree was stolen, but he still had enough trees left to plant exactly ten equal rows. If he bought the least number of trees satisfying these three conditions, how many trees did he buy?

ant101
Feb 6, 2018

#2**+2 **

The LCM of [8, 9, 10] = 360

360 - 8 =352 trees Jax bought

352/8 =44 trees

351/9=39 trees

350/10=35 trees.

Guest Feb 6, 2018

edited by
Guest
Feb 6, 2018

#4**+3 **

Melody: I wanted a number that was less than the LCM of 360, and that was divisible EVENLY by 8 and that ended in 2. Also, quickly realized that 352 - 1 =351 is divisible EVENLY by 9 since 3+5+1 =9. And finally, realized that 351 - 1=350 which is also divisible by 10 EVENLY !!.

Guest Feb 6, 2018

edited by
Guest
Feb 6, 2018

#6**+3 **

Solve this by setting up a system of modular equations.

\(\begin{array}{rcll} n &\equiv& {\color{red}0} \pmod {{\color{green}8}} \\ n &\equiv& {\color{red}1} \pmod {{\color{green}9}} \\ n &\equiv& {\color{red}2} \pmod {{\color{green}10}} \\ \text{Set } m &=& 8\cdot 9\cdot 10 = 720 \\ \end{array}\)

The first product zero— included as a formality.

Eurler totients calculated from non-prime numbers.

\(\small{ \begin{array}{l} n = {\color{red}0} \cdot {\color{green}9\cdot 10} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}9 \cdot 10) }^{\varphi({\color{green}8}) -1 } \pmod {{\color{green}8}} ] }_{=\text{modulo inverse }(9\cdot 10) \mod 8 } }_{=(9\cdot 10)^{4-1} \mod {8}} }_{=(9\cdot 10)^{3} \mod {8}} }_{=(90\pmod{8})^{3} \mod {8}} }_{=(2)^{3} \mod {8}} }_{= 0} + {\color{red}1} \cdot {\color{green}8\cdot 10} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}8\cdot 10) }^{\varphi({\color{green}9}) -1} \pmod {{\color{green}9}} ] }_{=\text{modulo inverse } (8\cdot 10) \mod {9}} }_{=(8\cdot 10)^{6-1} \mod {9}} }_{=(8\cdot 10)^{5} \mod {9}} }_{=(80\pmod{9})^{5} \mod {9}} }_{=(8)^{5} \mod {9}} }_{=8} + {\color{red}2} \cdot {\color{green}8\cdot 9} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ { (\color{green}8\cdot 9) }^{\varphi({\color{green}10}) -1 } \pmod {{\color{green}10}} ] }_{=\text{modulo inverse } (8\cdot 9) \mod 10 } }_{=(8\cdot 9)^{4-1} \mod { 10}} }_{=(8\cdot 9)^{3} \mod {10}} }_{=(72\pmod{10})^{3} \mod {10}} }_{=(2)^{3} \mod {10}} }_{=8}\\ \\ n = {\color{red}0} \cdot {\color{green}9\cdot 10} \cdot [0] + {\color{red}1} \cdot {\color{green}8\cdot 10} \cdot [8] + {\color{red}2} \cdot {\color{green}8\cdot 9} \cdot [8] \\ n = 0+ 640 + 1152 \\ n = 1792 \\\\ n \pmod {m}\\ = 1792 \pmod {720} \\ = 352 \\ \mathbf{n_{min}} \mathbf{=} \mathbf{352} \end{array} } \)

GingerAle
Feb 6, 2018

#8**0 **

Thanks Ginger,

I certainly like the way your working begins but I haven't worked out this Euler's totient yet.

I'll just take your word for it. :)

You should not be so critical of our guest.

There is nothing wrong with good intuition and sometimes the thought processes are very hard to explain.

Melody
Feb 6, 2018

#9**0 **

My criticism isn’t because of his intuition. It’s because of his lack of an explanation or the fluff and blarney he uses in its stead. His explanation of reasoning does not extend to similar problems nor does it really explain the solution method for this exact problem.

All mathamations develop evolving intuition in solving problems. This usually develops as the students learn the algorithms and rote formulas for solutions. For most, mathematics starts with “Ours is not to know the reason why, it’s to invert the divisor and multiply.”

While Mr. BB appears to have an intuition to solve this and other related modulo problems, his explanations never convey any substance for reason, nor foundation for a rote formula that is useful to anyone.

A case in point. You have a degree in mathematics and decades of practiced skill. You have a level intuition that far exceeds the vast majority of college students and most graduates. Yet, Mr. BB’s explanation didn’t trigger much, if any, understanding for the nature of the problem or its solution. If this doesn’t float any part of __your__ banana boat, then no one has a prayer to the banana goddess of hope for any understanding.

Though his comments may be true, they are only true by tautology. They are not very useful for this particular problem, and they are useless for any related problem. No one learns anything from Mr. BB’s Blarney, except how to become a Blarney Master. He is one too. . . . One of the best I’ve seen.

GA

GingerAle
Feb 7, 2018