Difference between revisions of "2006 AIME I Problems/Problem 3"

Latest revision as of 21:47, 13 October 2024

Problem

Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is $\frac{1}{29}$ of the original integer.

Solutions

Solution 1

Suppose the original number is $N = \overline{a_na_{n-1}\ldots a_1a_0},$ where the $a_i$ are digits and the first digit, $a_n,$ is nonzero. Then the number we create is $N_0 = \overline{a_{n-1}\ldots a_1a_0},$ so $N = 29N_0.$ But $N$ is $N_0$ with the digit $a_n$ added to the left, so $N = N_0 + a_n \cdot 10^n.$ Thus, $N_0 + a_n\cdot 10^n = 29N_0$ $a_n \cdot 10^n = 28N_0.$ The right-hand side of this equation is divisible by seven, so the left-hand side must also be divisible by seven. The number $10^n$ is never divisible by $7,$ so $a_n$ must be divisible by $7.$ But $a_n$ is a nonzero digit, so the only possibility is $a_n = 7.$ This gives $7 \cdot 10^n = 28N_0$ or $10^n = 4N_0.$ Now, we want to minimize both $n$ and $N_0,$ so we take $N_0 = 25$ and $n = 2.$ Then $N = 7 \cdot 10^2 + 25 = \boxed{725},$ and indeed, $725 = 29 \cdot 25.$ $\square$

Solution 2

Let $N$ be the required number, and $N'$ be $N$ with the first digit deleted. Now, we know that $N<1000$ (because this is an AIME problem). Thus, $N$ has $1,$ $2$ or $3$ digits. Checking the other cases, we see that it must have $3$ digits. Let $N=\overline{abc}$ , so $N=100a+10b+c$ . Thus, $N'=\overline{bc}=10b+c$ . By the constraints of the problem, we see that $N=29N'$ , so $100a+10b+c=29(10b+c).$ Now, we subtract and divide to get $100a=28(10b+c)$ $25a=70b+7c.$ Clearly, $c$ must be a multiple of $5$ because both $25a$ and $70b$ are multiples of $5$ . Thus, $c=5$ . Now, we plug that into the equation: $25a=70b+7(5)$ $25a=70b+35$ $5a=14b+7.$ By the same line of reasoning as earlier, $a=7$ . We again plug that into the equation to get $35=14b+7$ $b=2.$ Now, since $a=7$ , $b=2$ , and $c=5$ , our number $N=100a+10b+c=\boxed{725}$ .

Here's another way to finish using this solution. From the above, you have $100a = 28(10b + c).$ Divide by $4$ , and you get $25a = 7(10b + c).$ This means that $25a$ has to be divisible by $7$ , and hence $a = 7.$ Now, solve for $25 = 10b + c$ , which gives you $a = 7, b = 2, c = 5$ , giving you the number $\boxed{725}$

Solution 3 (Quick)

Note that if we let the last digit be $c$ we must have $9c \equiv c \pmod{10}.$ Thus we either have $c=0$ which we can quickly check to be impossible (since the number after digit removal could be 10,20,30) or $c=5.$ Testing 5, 15, and 25 as the numbers after removal we find that our answer is clearly $29 \cdot 25 = 725.$

~Dhillonr25

(Note that the quick checking of six numbers was possible thanks to AIME problems having answers less than 1000).

@@ Line 2: / Line 2: @@
 Find the least [[positive]] [[integer]] such that when its leftmost [[digit]] is deleted, the resulting integer is <math>\frac{1}{29}</math> of the original integer.
-== Solution ==
+== Solutions ==
-The number can be represented as <math>10^na+b</math>, where <math> a </math> is the leftmost digit, and <math> b </math> is the rest of the number. We know that <math>b=\frac{10^na+b}{29} \implies 28b=2^2\times7b=10^na</math>. Thus <math> a </math> has to be 7 since <math> 10^n </math> can not have 7 as a factor, and the smallest <math> 10^n </math> can be and have a factor of <math> 2^2 </math> is <math> 10^2=100. </math> We find that <math> b </math> is 25, so the number is 725.
+=== Solution 1 ===
+Suppose the original number is <math>N = \overline{a_na_{n-1}\ldots a_1a_0},</math> where the <math>a_i</math> are digits and the first digit, <math>a_n,</math> is nonzero. Then the number we create is <math>N_0 = \overline{a_{n-1}\ldots a_1a_0},</math> so <cmath>N = 29N_0.</cmath> But <math>N</math> is <math>N_0</math> with the digit <math>a_n</math> added to the left, so <math>N = N_0 + a_n \cdot 10^n.</math> Thus, <cmath>N_0 + a_n\cdot 10^n = 29N_0</cmath> <cmath>a_n \cdot 10^n = 28N_0.</cmath> The right-hand side of this equation is divisible by seven, so the left-hand side must also be divisible by seven. The number <math>10^n</math> is never divisible by <math>7,</math> so <math>a_n</math> must be divisible by <math>7.</math> But <math>a_n</math> is a nonzero digit, so the only possibility is <math>a_n = 7.</math> This gives <cmath>7 \cdot 10^n = 28N_0</cmath> or <cmath>10^n = 4N_0.</cmath> Now, we want to minimize ''both'' <math>n</math> and <math>N_0,</math> so we take <math>N_0 = 25</math> and <math>n = 2.</math> Then <cmath>N = 7 \cdot 10^2 + 25 = \boxed{725},</cmath> and indeed, <math>725 = 29 \cdot 25.</math> <math>\square</math>
+=== Solution 2 ===
+Let <math>N</math> be the required number, and <math>N'</math> be <math>N</math> with the first digit deleted. Now, we know that <math>N<1000</math> (because this is an AIME problem). Thus, <math>N</math> has <math>1,</math> <math>2</math> or <math>3</math> digits. Checking the other cases, we see that it must have <math>3</math> digits.
+Let <math>N=\overline{abc}</math>, so <math>N=100a+10b+c</math>. Thus, <math>N'=\overline{bc}=10b+c</math>. By the constraints of the problem, we see that <math>N=29N'</math>, so <cmath>100a+10b+c=29(10b+c).</cmath> Now, we subtract and divide to get <cmath>100a=28(10b+c)</cmath> <cmath>25a=70b+7c.</cmath> Clearly, <math>c</math> must be a multiple of <math>5</math> because both <math>25a</math> and <math>70b</math> are multiples of <math>5</math>. Thus, <math>c=5</math>. Now, we plug that into the equation: <cmath>25a=70b+7(5)</cmath> <cmath>25a=70b+35</cmath> <cmath>5a=14b+7.</cmath> By the same line of reasoning as earlier, <math>a=7</math>. We again plug that into the equation to get <cmath>35=14b+7</cmath> <cmath>b=2.</cmath> Now, since <math>a=7</math>, <math>b=2</math>, and <math>c=5</math>, our number <math>N=100a+10b+c=\boxed{725}</math>.
+Here's another way to finish using this solution. From the above, you have <cmath>100a = 28(10b + c).</cmath> Divide by <math>4</math>, and you get <cmath>25a = 7(10b + c).</cmath> This means that <math>25a</math> has to be divisible by <math>7</math>, and hence <math>a = 7.</math> Now, solve for <math>25 = 10b + c</math>, which gives you <math>a = 7, b = 2, c = 5</math>, giving you the number <math>\boxed{725}</math>
+== Solution 3 (Quick) ==
+Note that if we let the last digit be <math>c</math> we must have <math>9c \equiv c \pmod{10}.</math> Thus we either have <math>c=0</math> which we can quickly check to be impossible (since the number after digit removal could be 10,20,30) or <math>c=5.</math> Testing 5, 15, and 25 as the numbers after removal we find that our answer is clearly <math>29 \cdot 25 = 725.</math>
+~Dhillonr25
+(Note that the quick checking of six numbers was possible thanks to AIME problems having answers less than 1000).
 == See also ==
@@ Line 10: / Line 27: @@
 [[Category:Intermediate Number Theory Problems]]
+{{MAA Notice}}

2006 AIME I (Problems • Answer Key • Resources)
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