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Tens Digit of the Power Tower 7, 7^7, ... (t1000) | JEE

JEE Maths question with a full step-by-step solution.

Question

Let

t_1=7

t_2=7^7

t_3=7^{7^7}

, and so on. The digit at the tens place of

t_{1000}

8

0

6

4

correct

Solution

Step 1: The tens digit depends only on the last two digits. Since

7^4=2401

ends in

01

, the number

7^{4k+r}

ends in the same two digits as

7^r

; so the last two digits of a power of

7

depend on the exponent modulo

4

. Step 2: Now

t_{1000}=7^{\,t_{999}}

, so reduce

t_{999}\pmod 4

. As

7\equiv-1\pmod 4

and

t_{999}

has an odd exponent,

t_{999}\equiv(-1)^{\text{odd}}\equiv 3\pmod 4.

Step 3: Hence

t_{1000}

ends like

7^3=343

, i.e. last two digits

43

. Tens digit

=4.

Correct answer: (4)

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