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What is the smallest poistive integer n such that n^4 + 6n^3 + 11n^2 + 6n is divisible by 700?

 Nov 25, 2019
 #1
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n = 25

 

 25^4 + 6*25^3 + 11*25^2 + 6*25 =491,400 mod 700 =0

 Nov 25, 2019
 #2
avatar+2850 
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factor

 

n^4 + 6n^3 + 11n^2 + 6n

 

n(n^3 + 6n^2 + 11n + 6)

 

n(n^2(n+6)+11(n+6))

 

n(n2+11)(n+6)

 

And i leave up to u

 Nov 25, 2019
 #3
avatar+24407 
+2

What is the smallest positive integer \(n\) such that \(n^4 + 6n^3 + 11n^2 + 6n\) is divisible by \(700\)?

 

\(\begin{array}{|rcll|} \hline n^4 + 6n^3 + 11n^2 + 6n = n*(n+1)*(n+2)*(n+3) \\ \hline \end{array}\)

 

\(\begin{array}{|r|r|r|c|} \hline n& n*(n+1)*(n+2)*(n+3) \\ \hline 1& 1*2*3*4 & 24 & \\ 2& 2*3*4*5 & 120 & \\ 3& 3*4*5*6 & 360 & \\ 4& 4*5*6*7 & 840 & \\ 5& 5*6*7*8 & 1680 & \\ 6& 6*7*8*9 & 3024 & \\ 7& 7*8*9*10 & 5040 & \\ 8& 8*9*10*11 & 7920 & \\ 9& 9*10*11*12 & 11880 & \\ 10& 10*11*12*13 & 17160 & \\ 11& 11*12*13*14 & 24024 & \\ 12& 12*13*14*15 & 32760 & \\ 13& 13*14*15*16 & 43680 & \\ 14& 14*15*16*17 & 57120 & \\ 15& 15*16*17*18 & 73440 & \\ 16& 16*17*18*19 & 93024 & \\ 17& 17*18*19*20 & 116280 & \\ 18& 18*19*20*21 & 143640 & \\ 19& 19*20*21*22 & 175560 & \\ 20& 20*21*22*23 & 212520 & \\ 21& 21*22*23*24 & 255024 & \\ 22& 22*23*24*25 & 303600 & \\ 23& 23*24*25*26 & 358800 & \\ 24& 24*25*26*27 & 421200 & \\ \color{red}25& 25*26*27*28 & 491400 & \text{divisible by 700} \\ 26& 26*27*28*29 & 570024 & \\ 27& 27*28*29*30 & 657720 & \\ 28& 28*29*30*31 & 755160 & \\ 29& 29*30*31*32 & 863040 & \\ 30& 30*31*32*33 & 982080 & \\ 31& 31*32*33*34 & 1113024 & \\ 32& 32*33*34*35 & 1256640 & \\ 33& 33*34*35*36 & 1413720 & \\ 34& 34*35*36*37 & 1585080 & \\ 35& 35*36*37*38 & 1771560 & \\ 36& 36*37*38*39 & 1974024 & \\ 37& 37*38*39*40 & 2193360 & \\ 38& 38*39*40*41 & 2430480 & \\ 39& 39*40*41*42 & 2686320 & \\ 40& 40*41*42*43 & 2961840 & \\ 41& 41*42*43*44 & 3258024 & \\ 42& 42*43*44*45 & 3575880 & \\ 43& 43*44*45*46 & 3916440 & \\ 44& 44*45*46*47 & 4280760 & \\ 45& 45*46*47*48 & 4669920 & \\ 46& 46*47*48*49 & 5085024 & \\ 47& 47*48*49*50 & 5527200 & \text{divisible by 700} \\ 48& 48*49*50*51 & 5997600 & \text{divisible by 700} \\ 49& 49*50*51*52 & 6497400 & \text{divisible by 700} \\ 50& 50*51*52*53 & 7027800 & \\ \dots & \ldots & \ldots \\ \hline \end{array}\)

 

laugh

 Nov 25, 2019

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