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h0pe
 one year ago
There is a single sequence of integers \(a_2, a_3, a_4, a_5, a_6, a_7\) such that
\[\frac{5}{7} = \frac{a_2}{2!} + \frac{a_3}{3!} + \frac{a_4}{4!} + \frac{a_5}{5!} + \frac{a_6}{6!} + \frac{a_7}{7!},\]
and \(0 \le a_i < i\) for \(i = 2, 3, \dots 7\).
Find \(a_2 + a_3 + a_4 + a_5 + a_6 + a_7\).
h0pe
 one year ago
There is a single sequence of integers \(a_2, a_3, a_4, a_5, a_6, a_7\) such that \[\frac{5}{7} = \frac{a_2}{2!} + \frac{a_3}{3!} + \frac{a_4}{4!} + \frac{a_5}{5!} + \frac{a_6}{6!} + \frac{a_7}{7!},\] and \(0 \le a_i < i\) for \(i = 2, 3, \dots 7\). Find \(a_2 + a_3 + a_4 + a_5 + a_6 + a_7\).

This Question is Closed

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3use `\( latex mess goes here \)` for inline latex expressions

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3` \[ \] ` puts a new line at the start and end of expression

h0pe
 one year ago
Best ResponseYou've already chosen the best response.1It's getting late, so I'll fix it when I get on tomorrow.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3I fixed it for now, see if it still looks the same

freckles
 one year ago
Best ResponseYou've already chosen the best response.0* (bookmarking for tomorrow; sounds interesting to me)

h0pe
 one year ago
Best ResponseYou've already chosen the best response.1Thanks @ganeshie8 it looks so much cleaner

h0pe
 one year ago
Best ResponseYou've already chosen the best response.1I don't understand how you did that...

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3\[\frac{5}{7} = \frac{a_2}{2!} + \frac{a_3}{3!} + \frac{a_4}{4!} + \frac{a_5}{5!} + \frac{a_6}{6!} + \frac{a_7}{7!},\] multuply \(7!\) through out and get \[5\cdot 6! = 7\cdot 6\cdot 5\cdot 4\cdot 3a_2+7\cdot 6\cdot 5\cdot 4a_3+7\cdot 6\cdot 5a_4+7\cdot 6a_5+7a_6+a_7\] taking \(\mod 7\) both sides gives \[5(1)\equiv 0+a_7\pmod{7} \implies a_7 = 2\] taking \(\mod 6\) both sides gives \[0\equiv 0+a_6+a_7\pmod{6} \implies a_6 = 4\] see if you can find other values similarly

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3i have fixed a typo... please go thru again

h0pe
 one year ago
Best ResponseYou've already chosen the best response.1For 5 should I now do \[0≡0+a_5+a_6+a_7(mod 5)=a_5=4\] For 4: \[0\equiv0+a_4+a_5+a_6+a_7(mod4)=a_4=2\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3doesn't look correct

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3we have equation : \[5\cdot 6! = 7\cdot 6\cdot 5\cdot 4\cdot 3a_2+7\cdot 6\cdot 5\cdot 4a_3+7\cdot 6\cdot 5a_4+7\cdot 6a_5+7a_6+a_7\] taking mod5 should give \[0\equiv 0+7\cdot 6a_5 + 7a_6 + a_7 \pmod{5}\] plugin the previous known values and solve \(a_5\)

h0pe
 one year ago
Best ResponseYou've already chosen the best response.1Right, forgot that part. \[0\equiv6a_5+37 (\mod5)\] which is \[0\equiv55(\mod5)\] so \[a_5=3\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3i see lot of typoes/mistakes in ur reply

h0pe
 one year ago
Best ResponseYou've already chosen the best response.1So first \[0\equiv7*6a_5+28+2(mod 5)=0\equiv42a_5+30(mod 5)\] Then \[a_5=5\]

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3thats right, but can \(a_5\) be 5 ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3from hypothesis \(0\le a_5\lt 5\) right

h0pe
 one year ago
Best ResponseYou've already chosen the best response.1Oh, right. \(a_5\) has to be 0.

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.3Yes, try working others

h0pe
 one year ago
Best ResponseYou've already chosen the best response.1\[5⋅6!=7⋅6⋅5⋅4⋅3a_2+7⋅6⋅5⋅4a_3+7⋅6⋅5a_4+7⋅6a_5+7a_6+a_7\] How do you take mod 4

freckles
 one year ago
Best ResponseYou've already chosen the best response.0Anything that with a factor 4 will have a remainder of 0 when dividing the thing by 4. That is like for example 6!=5*4*3*2 so 6! mod 4 is 0

h0pe
 one year ago
Best ResponseYou've already chosen the best response.1so complicated ugh I'm going to write this huge thing down on paper
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