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{\bf NAME: (print!)}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
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 {\bf E-Mail address:}  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

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{\bf MATH 356, Dr. Z. ,  Exam I, Mon., Oct.  14, 2024
8:30-9:50am SEC-204}
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{\bf No Calculators! No Cheatsheets!}

Write the final answer to each problem in the space provided. Incorrect answers (even due to minor errors)
can receive at most one half partial credit, so please check and double-check your answers.

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1. \qquad  (out of 10)    
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2.  \qquad  (out of 10)    
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3.   \qquad (out of 10)    
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4.   \qquad (out of 10)    
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5.   \qquad (out of 10)    
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6.    \qquad (out of 10)    
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7.    \qquad (out of 10)    
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8.    \qquad (out of 10)    
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9.    \qquad (out of 10)    
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10.    \qquad (out of 10)    
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------------------
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tot.: \qquad \qquad (out of 100)    

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{\bf 1}. (10 pts.)  
Solve, if possible, the following linear congruences
$$ 
3x \equiv 8 \pmod {35}
$$


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{\bf Ans.}:  $x=$
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{\bf 2}. (10 pts.)   Find the two last (decimal) digits of $1013^{17}$
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{\bf Ans.}:  
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{\bf 3}. (10 pts.)  
Find out whether it is possible to express $1$ as a linear combination 
$$
1=m \cdot 44 +n \cdot 91 \quad ,
$$ 
for some integers $m$ and $n$, and if it is possible, find it.

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{\bf Ans.}:  $m=$ \quad \quad $n=$ \quad \quad \quad .
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{\bf 4}. (10 pts.)  Find  the set of divisors of $105$

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{\bf Ans.}:  $Div(105)=\{ \qquad  \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad  \}$
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{\bf 5}. (10 pts.)  Express $25025$ as a product of prime powers.

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{\bf Ans.}:  
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{\bf 6}. (10 pts.)  

Recall that the {\it Euclid-Mullin sequence}  is defined by
$M_1=2$, and for $n>1$, $M_n$ is the smallest prime divisor of $M_1 \cdot M_2 \cdots M_{n-1}+1$.

Find the first {\bf five} terms of the Euclid-Mullin sequence.

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{\bf Ans.}:   $M_1=2$ \qquad ,  \qquad 
$M_2=\qquad$ , \qquad $M_3=\qquad$,  
\qquad $M_4=\qquad$, \qquad  $M_5=\qquad$ \qquad .
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{\bf 7}. (10 pts.)  
Express the integer $301$ (writen in our usual (base 10) notation)
in  base $5$, in (i) sparse notation (4 pts) (ii) dense notation (3 pts) (iii) base-five positional notation (3 pts)

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{\bf Ans.}:  (i)
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(ii)
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(iii)
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{\bf 8}. (10 pts.)  
Use Karatsuba's algorithm (no credit for other methods!), to find
the product
$$
99 \cdot 53 \quad .
$$
{\bf Reminder}: $(10a+b)(10c+d)= 100ac+ 10( (a+b)(c+d)-ac-bd))+ bd$ .

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{\bf Ans.}:  
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{\bf 9}. (10 pts.)  
Let $F_n$ be the Fibonacci numbers.
Give a Zeilberger-style proof of the following identity, using $N_0=3$, of the identity
$$
\sum_{i=1}^{2n-1} F_i F_{i+1} = F_{2n}^2 \quad .
$$

{\bf Disclaimer:} To get a fullly rigorous proof, we need $N_0=5$, but I am being nice, so I only ask you to do
it for $n=1,2,3$, so this is a semi-rigorous proof.

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{\bf 10}. (10 pts.)  

Use the Fundamental Theorem of Discrete Calculus  to prove
$$
\sum_{i=1}^{n} 3^i = \frac{3}{2}(3^{n}-1) \quad .
$$

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