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[et_pb_section fb_built="1" _builder_version="4.2.2" custom_padding="||42px|||"][et_pb_row _builder_version="4.2.2" module_alignment="center"][et_pb_column type="4_4" _builder_version="4.2.2"][et_pb_blurb _builder_version="4.2.2"][/et_pb_blurb][et_pb_blurb _builder_version="4.2.2"][/et_pb_blurb][et_pb_team_member name="American Mathematics contest 10 (AMC 10) - Number Theory problems " _builder_version="4.2.2" header_level="h1" header_font="Acme||||||||" header_text_align="center" header_font_size="27px" body_font="Acme||||||||" body_text_align="center" body_letter_spacing="1px" background_color="#032b35" background_layout="dark" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"] Try these AMC 10 Number Theory Questions and check your knowledge![/et_pb_team_member][et_pb_image src="https://www.staging18.cheenta.com/wp-content/uploads/2020/01/AMC-8-Geometry-Questions-10.png" _builder_version="4.2.2"][/et_pb_image][et_pb_team_member name="AMC 10A, 2020, Problem 4" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" hover_enabled="0" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $0.50$ per mile, and her only expense is gasoline at $2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?

$\textbf{(A) }20 \qquad\textbf{(B) }22 \qquad\textbf{(C) }24 \qquad\textbf{(D) } 25\qquad\textbf{(E) } 26$
[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 6" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" hover_enabled="0" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

$\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$
[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 7" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" hover_enabled="0" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

$\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$
[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 8" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" hover_enabled="0" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the value of

$1+2+3-4+5+6+7-8+\cdots+197+198+199-200?$

$\textbf{(A) } 9,800 \qquad \textbf{(B) } 9,900 \qquad \textbf{(C) } 10,000 \qquad \textbf{(D) } 10,100 \qquad \textbf{(E) } 10,200$
[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 9" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" hover_enabled="0" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$

$\textbf{(A) } 9 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 27 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 77$
[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 17" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" hover_enabled="0" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Define\[P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).\]How many integers $n$ are there such that $P(n)\leq 0$?

$\textbf{(A) } 4900 \qquad \textbf{(B) } 4950\qquad \textbf{(C) } 5000\qquad \textbf{(D) } 5050 \qquad \textbf{(E) } 5100$
[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 7 21" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" hover_enabled="0" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < … < a_k$ such that\[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.\]What is $k?$

$\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$
[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 22" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" hover_enabled="0" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]For how many positive integers $n \le 1000$ is\[\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor\]not divisible by $3$? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

$\textbf{(A) } 22 \qquad\textbf{(B) } 23 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 26$
[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 24" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" hover_enabled="0" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Let $n$ be the least positive integer greater than $1000$ for which\[\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.\]What is the sum of the digits of $n$?

$\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 21\qquad\textbf{(E) } 24$
[/et_pb_team_member][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built="1" _builder_version="4.2.2"][et_pb_row _builder_version="4.2.2"][et_pb_column type="4_4" _builder_version="4.2.2"][/et_pb_column][/et_pb_row][/et_pb_section]

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