2021 AMC 10B Fall Problem 19

Question

Let

N

be the positive integer

7777\ldots777,

a

313

-digit number where each digit is a

7.

Let

f(r)

be the leading digit of the

r

th root of

N.

What is

\begin{aligned} &f(2)+f(3)+f(4)\\ &\quad{}+f(5)+f(6)? \end{aligned}

8

9

11

22

29

Answer: A

Concepts:digits exponent bounding to limit cases

Difficulty rating: 1990

Small hint:

Bound

N

between

7\cdot10^{312}

and

8\cdot10^{312}

Big hint:

For each root, factor out the largest power of

10

Solution:

The number

N

satisfies

7\cdot10^{312}\lt N\lt 8\cdot10^{312}.

Multiplying or dividing by a power of

10

only shifts the decimal point, so we only need the leading factor left after taking out the largest convenient power of

10.

For

r=2,

\sqrt N

has leading factor between

\sqrt7

and

\sqrt8,

so

f(2)=2.

For

r=3,

the leading factor is between

\sqrt[3]{7}

and

\sqrt[3]{8},

so

f(3)=1.

For

r=4,

the leading factor is between

\sqrt[4]{7}

and

\sqrt[4]{8},

so

f(4)=1.

For

r=5,

since

312=5\cdot62+2,

the leading factor is between

\sqrt[5]{700}

and

\sqrt[5]{800}.

Since

3^5\lt700\lt800\lt4^5,

f(5)=3.

For

r=6,

the leading factor is between

\sqrt[6]{7}

and

\sqrt[6]{8},

so

f(6)=1.

The sum is

2+1+1+3+1=8.

Thus, the answer is A .

2021 AMC 10B Fall Problem 19

Answer: A

Difficulty rating: 1990

Small hint:

Big hint:

Solution:

Problem 19 in other years