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2022 AMC 12B Problems/Problem 15

Revision as of 02:02, 18 November 2022 by Arjken (talk | contribs)

Problem: One of the following numbers is not divisible by any prime number less than 10. Which is it? $\text{(A)} 2^{606}-1 \text{(B)} 2^{606}+1 \text{(C)} 2^{607}-1 \text{(D)} 2^{607}+1 \text{(E)} 2^{607}+3^{607}$

Solution 1 (Process of Elimination)

We examine option E first. $2^{607}$ has a units digit of $8$ (Taking the units digit of the first few powers of two gives a pattern of $2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6,\cdots$) and $3^{607}$ has a units digit of $7$ (Taking the units digit of the first few powers of three gives a pattern of $3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, 1,\cdots$). Adding $7$ and $8$ together, we get $15$, which is a multiple of $5$, meaning that $2^{607}+3^{607}$ is divisible by 5.

Next, we examine option D. We take the first few powers of $2$ added with $1$: \[2^1+1=3\] \[2^2+1=5\] \[2^3+1=9\] \[2^4+1=17\] \[2^5+1=33\] \[2^6+1=65\] \[2^7+1=129\]

We see that the odd powers of $2$ added with 1 are multiples of three. If we continue this pattern, $2^{607}+1$ will be divisible by $3$. (The reason why this pattern works: When you multiply $2 \equiv2\text{mod} 3$ by $2$, you obtain $4 \equiv1 \text{mod} 3$. Multiplying by $2$ again, we get $1\cdot2\equiv2 \text{mod} 3$. We see that in every cycle of two powers of $2$, it goes from $2 \text{mod}3$ to $1 \text{mod}3$ and back to $2 \text{mod}3$.)

Next, we examine option B. We see that $2^{606}$ has a units of digits of $4$ (Taking the units digit of the first few powers of two gives a pattern of $2, 4, 8, 6, 2, 4, 8, 6, 2, 4, 8, 6,\cdots$). Adding $1$ to $4$, we get $5$. Since $2^{606}+1$ has a units digit of $5$, it is divisible by $5$.

Lastly, we examine $2^{606}+1$. Using the sum of cubes factorization $a^3+b^3=(a+b)(a^2-ab+b^2)$, we have $2^{606}+1^3=(2^{202}+1)(2^{404}+2^{202}+1)$. Since $2^{202}$ ends with a $4$, and $4+1=5$, $2^{606}+1$ is a multiple of $5$, which means it is divisible by $5$.

Since we have eliminated every option except one, $\boxed{\text{(C)}2^{607}-1}$ is not divisible by any prime less than $10$.

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