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Difference between revisions of "1999 CEMC Pascal Problems/Problem 20"

(Created page with "==Problem== The numbers <math>49, 29, 9, 40, 22, 15, 53, 33, 13, 47</math> are grouped in pairs so that the sum of each pair is the same. Which number is paired with <math>15<...")
 
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==Problem==
 
==Problem==
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The numbers <math>49, 29, 9, 40, 22, 15, 53, 33, 13, 47</math> are grouped in pairs so that the sum of each pair is the same. Which number is paired with <math>15</math>?
+
The units (ones) digit in the product <math>(5 + 1)(5^3 + 1)(5^6 + 1)(5^{12} + 1)</math> is
  
 
<math> \text{ (A) }\ 6 \qquad\text{ (B) }\ 5 \qquad\text{ (C) }\ 2 \qquad\text{ (D) }\ 1 \qquad\text{ (E) }\ 0</math>
 
<math> \text{ (A) }\ 6 \qquad\text{ (B) }\ 5 \qquad\text{ (C) }\ 2 \qquad\text{ (D) }\ 1 \qquad\text{ (E) }\ 0</math>

Latest revision as of 23:52, 30 June 2025

Problem

The units (ones) digit in the product $(5 + 1)(5^3 + 1)(5^6 + 1)(5^{12} + 1)$ is

$\text{ (A) }\ 6 \qquad\text{ (B) }\ 5 \qquad\text{ (C) }\ 2 \qquad\text{ (D) }\ 1 \qquad\text{ (E) }\ 0$

Solution 1

We can notice that the units digit of $5$ raised to any power is $5$ since $5 \times 5 = 25$, which has a units digit of $5$.

Thus, the units digit of the expression given is the same as the units digit of:

$(5 + 1)(5 + 1)(5 + 1)(5 + 1) = 6^4$

We can also notice that the units digit of $6$ to any power is $\boxed {\textbf {(A) } 6}$ using similar logic with $6 \times 6$ having a units digit of $6$.

~anabel.disher

Solution 1.1

We can use the same logic to find that the units digit of the expression given is the same as the units digit of $6^4$.

However, we can solve $6^4$ to get $1296$, which has a units digit of $\boxed {\textbf {(A) } 6}$.

~anabel.disher

Solution 2 (unrecommended)

We can solve for the value of the expression, and then use its ones digit: $(5 + 1)(5^3 + 1)(5^6 + 1)(5^{12} + 1)$

$=6 \times (125 + 1) \times (15625 + 1) \times (244140625 + 1)$

$=6 \times 126 \times 15626 \times 244140626$

$=2,884,095,714,938,256$

We then can see that the units digit is $\boxed {\textbf {(A) } 6}$.

However, this method is not recommended, and takes a lot of computation.

~anabel.disher

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