Question: What are the last two digits of 7²⁰⁰⁸?

A. 21
B. 61
C. 01
D. 41
E. 81

Answer: C

Approach Solution (1):
See 7¹=7 last two digit =07
7²=49 last two digit=49
7³=243 last two digit=43
7⁴=1701 last two digit=01
7⁵=8507 last two digit=07
7⁶=59549 last two digit=49
7⁷= last two digit=43
7⁸= last two digit=01
We notice that
last two digit in 7ⁿ=07 when n=4k+3
last two digit in 7ⁿ=49 when n=4k+2
last two digit in 7ⁿ=43 when n=4k+1
last two digit in 7ⁿ=01 when n=4k
2008=4*502=>7²⁰⁰⁸=7ⁿ=>n=504
Hence last two digit in 7²⁰⁰⁸ is 01
Correct option: C

Approach Solution (2):
7²⁰⁰⁸ can be written as (7⁴) ⁵⁰².
7⁴ = 2401, as we are interested in finding the last digit, so take only 01 of 2401,now the question reduces to last two digits of (01)⁵⁰².
When number ends with 1, its last digit is always 1, for second last digit, multiply 0 with last digit of power i.e 2
So last two digits = 01
Correct option: C

Approach Solution (3):
For finding out the last two digits of an odd number raised to a power, we should first try and reduce the base to a number ending in 1.
After that, we can use the property
Last two digits of (...a1)^(...b) will be [Last digit of a*b]1
Let us try and apply this concept in the given question
7²⁰⁰⁸
= 49¹⁰⁰⁴
= 2401⁵⁰²
= [Last digit of 0*2]1{Here I have used the concept mentioned above}
= 01
Correct option: C

“What are the last two digits of?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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