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What is the remainder when (a + 2)^5 + (a + 3)^4 + (a + 4)^3 + (a + 5)

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What is the remainder when (a + 2)^5 + (a + 3)^4 + (a + 4)^3 + (a + 5) [#permalink] New post  02 Apr 2020, 05:03
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What is the remainder when \((a + 2)^5 + (a + 3)^4 + (a + 4)^3 + (a + 5)^2 + (a + 6)\) is divided by \((a + 3)\), for any 2 digit number \(a\)?

A. 0
B. 1
C. 3
D. 5
E. 7
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What is the remainder when (a + 2)^5 + (a + 3)^4 + (a + 4)^3 + (a + 5) [#permalink] New post  02 Apr 2020, 05:23
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Dillesh4096 wrote:
What is the remainder when \((a + 2)^5 + (a + 3)^4 + (a + 4)^3 + (a + 5)^2 + (a + 6)\) is divided by \((a + 3)\), for any 2 digit number \(a\)?

A. 0
B. 1
C. 3
D. 5
E. 7


When (a + 2) is divided by (a+3) then remainder is -1

i.e. When \((a + 2)^5\) is divided by (a+3) then remainder is \((-1)^5 = -1\)

i.e. When \((a + 3)^4\) is divided by (a+3) then remainder is \(0\)

i.e. When \((a + 4)^3\) is divided by (a+3) then remainder is \((1)^3 = 1\)

i.e. When \((a + 5)^2\) is divided by (a+3) then remainder is \((2)^2 = 4\)

i.e. When \((a + 6)\) is divided by (a+3) then remainder is \(3\)

Final Remainder = -1+0+1+4+3 = 7

Answer: Option E

The concept video of remainder theorem is as follows:

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Re: What is the remainder when (a + 2)^5 + (a + 3)^4 + (a + 4)^3 + (a + 5) [#permalink] New post  02 Apr 2020, 06:09
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Dillesh4096 wrote:
What is the remainder when \((a + 2)^5 + (a + 3)^4 + (a + 4)^3 + (a + 5)^2 + (a + 6)\) is divided by \((a + 3)\), for any 2 digit number \(a\)?

A. 0
B. 1
C. 3
D. 5
E. 7


Solution:


Since, we can see that there are no options such as "Cannot be determined", it would be safe to assume that the answer will be same for any 2-digit number "a"
    o So, let's assume a as 17. (Since a + 3 = 20 and it may make our calculation easy).
      o Now, \(a + 2 = 19\)
         Remainder when \(19^5\) is divided by \(20 = -1^5 = -1\)
      o Now, \(a+ 3 = 20\)
         Remainder when \(20^4\) is divided by \(20 = 0\)
      o Now, \(a + 4 = 21\)
         Remainder when \(21^3\) is divided by \(20 = 1^3 = 1\)
      o Now, \(a + 5 = 22\)
         Remainder when \(22^2\) is divided \(20 = 2^2 = 4\)
      o Now, \(a + 6 = 23\)
         Remainder when 23 is divided \(20 = 3\)
      o Remainder when given expression is divided by \(a + 3 = -1 + 0 +1 +4+3 = 7\).
Hence, the correct answer is Option E.
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Re: What is the remainder when (a + 2)^5 + (a + 3)^4 + (a + 4)^3 + (a + 5) [#permalink] New post  02 Apr 2020, 06:55
[(a+2)^5+(a+3)^4+(a+4)^3+(a+5)^2+(a+6)]/ (a+3)
or, (a+2)^5/ (a+3) +(a+3)^4/ (a+3)+(a+4)^3/ (a+3)+(a+5)^2/ (a+3)+(a+6)/ (a+3)
now, 1st term will give -1 rem
2nd term will give 0 rem
3rd term will give +1 rem
4th term will give (+2)^2 rem
5th term will give +3 rem

so the actual remainder will be 4+3 =7

correct answer will be E.
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Re: What is the remainder when (a + 2)^5 + (a + 3)^4 + (a + 4)^3 + (a + 5) [#permalink]
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