18 is a common factor

Algebra Level 4

Find the remainder of

1 8 7 + ( 7 ) ( 1 8 6 ) + ( 21 ) ( 1 8 5 ) + ( 35 ) ( 1 8 4 ) + ( 35 ) ( 1 8 3 ) + ( 21 ) ( 1 8 2 ) + 7 ( 18 ) + 18 19 \large{\frac{18^7 + (7)(18^6) + (21)(18^5) +(35)(18^4) + (35)(18^3) + (21)(18^2) +7(18) +18}{19}} .


The answer is 17.

Paul Ryan Longhas by Paul Ryan Longhas , 24, Philippines

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2 solutions

Note that the expression is can be written as ( 18 + 1 ) 7 + 17 18 + 1 \frac{(18+1)^7 + 17}{18+1} .

Thus,

⇒ ( 18 + 1 ) 6 + 17 19 ⇒ \Rightarrow (18+1)^6 + \frac{17}{19} \Rightarrow the remainder is 17 17 .

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A better way to explain this is to show that 1, 7, 21, 35, 35, 21, 7 are elements in the 7th row of the Pascal Triangle.

Pi Han Goh - 5 years, 9 months ago
John Lesteя Tan
Sep 4, 2015
  • We can let x= 18
  • (x^7 + 7x^6+21x^5+35x^4+35x^3 +21x^2+7x+18) / (x+1)
  • then you can use factor theorem
  • x+1 = x-c
  • c = -1
  • so substitute -1 to the equation
  • -1 + 7 - 21 + 35 - 35 +21 - 7 + 18 = 17
  • so the remainder is 17
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