Consider the sequence: a1 = 101, a2 = 10101 , a3

Question

Consider the sequence: a1 = 101, a2 = 10101 , a3 = 1010101, and so on. Then ak is a composite number :(A) Iff k>=2 and 10k+1 +1 | 11(B) Iff k>=2 and 10k+1 -1 | 11(C) Iff k>=2 and k - 2 | 3(D) Iff k>=2

Arun · Accepted Answer

a) is wrong
Consider k = 3.
10^(k+1) + 1 = 10^4 + 1 = 10001
Clearly, 11 does NOT divide 10001 since 10001/11 = 909.18
But, a3 = 1010101 is composite since it is divisible by 101.

(b) is wrong
Consider k = 2.
10^(k+1) - 1 = 10^3 - 1 = 999
Clearly, 11 does NOT divide 999 since 999/11 = 90.82
But, a2 = 10101 is composite since it is divisible by 3.

(c) is wrong
Consider k = 4.
k - 2 = 4 - 2 = 2, which is NOT divisible by 3.
But, a4 = 101010101 is composite since it is divisible by 41.

So, (d) is your only choice left, even though it might be a bit hard to prove.

Algebra> Consider the sequence: a1 = 101, a2 = 101...

Consider the sequence: a1 = 101, a2 = 10101 , a3 = 1010101, and so on. Then a_k is a composite number :
(A) Iff k>=2 and 10^k+1+1 | 11
(B) Iff k>=2 and 10^k+1 -1 | 11
(C) Iff k>=2 and k - 2 | 3
(D) Iff k>=2

Nandish Panchal , 6 Years ago

Arun

Deepanjan Das

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Algebra> Consider the sequence: a1 = 101, a2 = 101...

Consider the sequence: a1 = 101, a2 = 10101 , a3 = 1010101, and so on. Then ak is a composite number :(A) Iff k>=2 and 10k+1 +1 | 11(B) Iff k>=2 and 10k+1 -1 | 11(C) Iff k>=2 and k - 2 | 3(D) Iff k>=2

Nandish Panchal , 6 Years ago

Arun

Deepanjan Das

Provide a better Answer & Earn Cool Goodies

LIVE ONLINE CLASSES

Ask a Doubt

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Consider the sequence: a1 = 101, a2 = 10101 , a3 = 1010101, and so on. Then a_k is a composite number :
(A) Iff k>=2 and 10^k+1+1 | 11
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(C) Iff k>=2 and k - 2 | 3
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