Category Archives: Number Theory

Number Theory Challenge – 1

Posted on November 4, 2023 by kevin

Let $n=4k+3$ is a prime number and $a^2+b^2\equiv 0\pmod{n}$, prove that $a\equiv b \equiv 0\pmod{n}$. Click for the solution Proof: If $a\equiv 0\pmod{n}$, because $a^2+b^2\equiv 0\pmod{n}$, $b\equiv 0\pmod{n}$. If $a\not\equiv 0\pmod{n}$, because $n$ is a prime number, according to Fermat’s … Continue reading →

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MathCounts Training – Number Theory – 5

Posted on September 15, 2023 by kevin

________ Let $(a\times b\times c)\div(a+b+c)=341$ to be an equation where $a$, $b$ and $c$ are consecutive positive integers. What is the least possible value of $a$? $$ $$ ________ The letters $A$, $B$, $C$, $D$, $E$ and $F$ represent digits … Continue reading →

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MathCounts Training – Number Theory – 4

Posted on September 8, 2023 by kevin

________ What is the greatest positive integer that must divide the sum of the first ten terms of any arithmetic sequence whose terms are positive integers?$$ $$ ________ A digit can be placed in each of the boxes for hundreds … Continue reading →

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MathCounts Training – Number Theory – 3

Posted on September 1, 2023 by kevin

________ What is the sum of all integer values of $n$ such that $\dfrac{20}{2n-1}$ is an integer? ________ What is the units digit of the sum of the sum of all the integers from 100 to 202 inclusive? ________ How … Continue reading →

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MathCounts Training – Number Theory – 2

Posted on August 25, 2023 by kevin

________ In base $b$, $441_b$ is equal to $n^2$ in base 10, and $351_b$ is equal to $(n-2)^2$. What is the value of $b$, expressed in base 10? ________ The base-three representation of $0.\overline{12}$ is equivalent to what base-ten common … Continue reading →

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