Need help - I can’t remember how to solve this type!

[Guest]

A man has seven sons, each is married, and wants to give away his prized marble collection.

Oldest son gets to pick as many as he wants, and his wife gets 1/9 of what is left.

2nd son gets to pick as many as the oldest son + 1, wife gets 1/9 of what is left.

3rd son gets to pick as many as 2nd son + 1, wife gets 1/9 of what’s left.

This continues to the last son, but there is nothing left so the wife gets zero.

How many marbles did the old man have, and how many did each wife get?

[Guest]

suppose the first son picks [tex]x[/tex] marbles and there are [tex]n[/tex] marbles in total. Then the others pick marbles as follows:

son1 [tex]=x[/tex], wife1 [tex]=\frac{n-x}{9}[/tex], remaining [tex]=\frac{8(n-x)}{9}[/tex]

son2 [tex]=x+1[/tex], wife2 [tex]=\displaystyle\frac{\displaystyle\frac{8(n-x)}{9}-(x+1)}{9}=\frac{8n-17x-9}{81}[/tex], remaining [tex]=\frac{8(8n-17x-9)}{81}[/tex]

son3 [tex]=x+2[/tex], wife3 [tex]=\displaystyle\frac{\displaystyle\frac{8(8n-17x-9)}{81}-(x+2)}{9}=\frac{64n-217x-234}{729}[/tex], remaining [tex]=\frac{8(64n-217x-234)}{729}[/tex]

son4 [tex]=x+3[/tex], wife4 [tex]=\displaystyle\frac{\displaystyle\frac{8(64n-217x-234)}{729}-(x+3)}{9}=\frac{512n-2465x-4059}{6561}[/tex], remaining [tex]=\frac{8(512n-2465x-4059)}{6561}[/tex]

son5 [tex]=x+4[/tex], wife5 [tex]=\displaystyle\frac{\displaystyle\frac{8(512n-2465x-4059)}{6561}-(x+4)}{9}=\frac{4096n-26281x-58716}{59049}[/tex], remaining [tex]=\frac{8(4096n-26281x-58716)}{59049}[/tex]

son6 [tex]=x+5[/tex], wife6 [tex]=\displaystyle\frac{\displaystyle\frac{8(4096n-26281x-58716)}{59049}-(x+5)}{9}=\frac{32768n-269297x-764973}{531441}[/tex], remaining [tex]=\frac{8(32768n-269297x-764973)}{531441}[/tex]

son7 [tex]=x+6[/tex], wife7 [tex]=\displaystyle\frac{\displaystyle\frac{8(32768n-269297x-764973)}{531441}-(x+5)}{9}=\frac{262144n-2685817x-8776989}{4782969}[/tex]

since the last wife doesn't get any, [tex]262144n-2685817x-8776989=0[/tex]

[tex]\Longrightarrow n=\frac{2685817x+8776989}{262144}\Longrightarrow n=10x+33+\frac{64377x+126237}{262144}[/tex]

so we need [tex]64377x+126237[/tex] to be a multiple of [tex]262144[/tex]. let the smallest multiple that satisfies this be [tex]262144a[/tex]. So,
[tex]x=\frac{262144a-126237}{64377}[/tex]. Now, [tex](262144a-126237)\mod64377=4636a-61860[/tex].

so we need [tex]4636a-61860[/tex] to be a multiple of [tex]64377[/tex]. let the smallest multiple that satisfies this be [tex]64377b[/tex]. So,
[tex]a=\frac{64377b+61860}{4636}[/tex]. Now, [tex](64377b+61860)\mod4636=4109b+1592[/tex].

so we need [tex]4109b+1592[/tex] to be a multiple of [tex]4636[/tex]. let the smallest multiple that satisfies this be [tex]4636c[/tex]. So,
[tex]b=\frac{4636c-1592}{4109}[/tex]. Now, [tex](4636c-1592)\mod4109=527c-1592[/tex].

so we need [tex]527c-1592[/tex] to be a multiple of [tex]4109[/tex]. let the smallest multiple that satisfies this be [tex]4109d[/tex]. So,
[tex]c=\frac{4109d+1592}{527}[/tex]. Now, [tex](4109d+1592)\mod527=420d+11[/tex].

so we need [tex]420d+11[/tex] to be a multiple of [tex]527[/tex]. let the smallest multiple that satisfies this be [tex]527e[/tex]. So,
[tex]d=\frac{527e-11}{420}[/tex]. Now, [tex](527e-11)\mod420=107e-11[/tex].

so we need [tex]107e-11[/tex] to be a multiple of [tex]420[/tex]. let the smallest multiple that satisfies this be [tex]420f[/tex]. So,
[tex]e=\frac{420f+11}{107}[/tex]. Now, [tex](420f+11)\mod107=99f+11[/tex].

so we need [tex]99f+11[/tex] to be a multiple of [tex]107[/tex]. let the smallest multiple that satisfies this be [tex]107g[/tex]. So,
[tex]f=\frac{107g-11}{99}[/tex]. Now, [tex](107g-11)\mod99=8g-11[/tex].

so we need [tex]8g-11[/tex] to be a multiple of [tex]99[/tex]. let the smallest multiple that satisfies this be [tex]99h[/tex]. So,
[tex]g=\frac{99h+11}{8}[/tex]. Now, [tex](99h+11)\mod8=3h+3[/tex].

so we need [tex]3h+3[/tex] to be a multiple of [tex]8[/tex]. let the smallest multiple that satisfies this be [tex]8i[/tex]. So,
[tex]h=\frac{8i-3}{3}[/tex]. Now, [tex](8i-3)\mod3=2i[/tex].

so we need [tex]2i[/tex] to be a multiple of [tex]3[/tex]. the smallest multiple that satisfies this is [tex]6\Longrightarrow i=3[/tex].

Now, working backward, [tex]h=\frac{8i-3}{3}=7\Longrightarrow g=\frac{99h+11}{8}=88\Longrightarrow f=\frac{107g-11}{99}=95\Longrightarrow e=\frac{420f+11}{107}=373\Longrightarrow d=\frac{527e-11}{420}=468\Longrightarrow c=\frac{4109d+1592}{527}=3652\Longrightarrow b=\frac{4636c-1592}{4109}=4120\Longrightarrow a=\frac{64377b+61860}{4636}=57225[/tex]

[tex]\Longrightarrow x=\frac{262144a-126237}{64377}=\boxed{233019}\Longrightarrow n=\frac{2685817x+8776989}{262144}=\boxed{2387448}[/tex]

using the above result, you can calculate how many marbles each person gets

[Guest]

THERE IS A MISTAKE IN THE ABOVE SOLUTION IN THE AMOUNT THE SEVENTH WIFE RECEIVES. HERE IS THE CORRECTED SOLUTION

son7 [tex]=x+6[/tex], wife7 [tex]=\displaystyle\frac{\displaystyle\frac{8(32768n-269297x-764973)}{531441}-(x+6)}{9}=\frac{262144n-2685817x-9308430}{4782969}[/tex]

since the last wife doesn't get any, [tex]262144n-2685817x-9308430=0[/tex]

[tex]\Longrightarrow n=\frac{2685817x+9308430}{262144}[/tex]. Now, [tex](2685817x+9308430)\mod262144=64377a-133390[/tex].

so we need [tex]64377x+133390[/tex] to be a multiple of [tex]262144[/tex]. let the smallest multiple that satisfies this be [tex]262144a[/tex]. So,
[tex]x=\frac{262144a-133390}{64377}[/tex]. Now, [tex](262144a-133390)\mod64377=4636a-4636[/tex].

so we need [tex]4636a-4636[/tex] to be a multiple of [tex]64377[/tex]. let the smallest multiple that satisfies this be [tex]64377b[/tex]. So,
[tex]a=\frac{64377b+4636}{4636}[/tex]. Now, [tex](64377b+4636)\mod4636=4109b[/tex].

so we need [tex]4109b[/tex] to be a multiple of [tex]4636[/tex]. the smallest multiple that satisfies this is [tex]0\Longrightarrow b=0[/tex].

Now, working backward, [tex]\Longrightarrow a=\frac{64377b+4636}{4636}=1[/tex]

[tex]\Longrightarrow x=\frac{262144a-133390}{64377}=\boxed{2}\Longrightarrow n=\frac{2685817x+8776989}{262144}=\boxed{56}[/tex]

using the above result, you can calculate how many marbles each person gets[/quote]