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The function f(x) is defined as x - 10[x/10]. ([x] is the greatest integer less than or equal to x). What is f(7) + f(7

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The function f(x) is defined as x - 10[x/10]. ([x] is the greatest integer less than or equal to x). What is f(7) + f(7

by Max@Math Revolution » Sun Mar 22, 2020 6:50 pm

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[GMAT math practice question]

The function f(x) is defined as x - 10[x/10]. ([x] is the greatest integer less than or equal to x). What is f(7) + f(7^2) + f(7^3) +…+ f(7^2020)?

A. 10036
B. 10039
C. 10040
D. 10100
E. 10107
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Re: The function f(x) is defined as x - 10[x/10]. ([x] is the greatest integer less than or equal to x). What is f(7) +

by Max@Math Revolution » Sun Mar 22, 2020 6:51 pm
=>

f(x) = x – 10[x/10] means the remainder of x when x is divided by 10.
For example, f(1234) = 1234 – 10*[1234/10] = 1234 – 10*[123.4] = 1234 – 10*123 = 1234 – 1230 = 4.

Units digit of Powers of base 7 repeats as follows.
f(7^1) = 7 ~ 7^5 ~ 7^9 ~ …
f(7^2) = f(49) = 9 ~ 7^6 ~ 7^10 ~ …
f(7^3) = f(343) = 3 ~ 7^7 ~ 7^11 ~ …
f(7^4) = f(2401) = 1 ~ 7^8 ~ 7^12 ~ …
f(7) + f(7^2) + f(7^3) +…+ f(7^2020)
= f(7) + f(7^2) + f(7^3) + f(7^4) + f(7^5) +…+ f(7^2017) + f(7^2018) + f(7^2019) + f(7^2020)
= (7 + 9 + 3 + 1) + (7 + 9 + 3 + 1) +…+ (7 + 9 + 3 + 1)
= 20 * 2020/4 = 20*505 = 10100

Therefore, D is the answer.
Answer: D
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