+0

# 2nd Help!

0
220
2
+363

Suppose a is an integer such that $$0 \le a \le 14$$ , and $$235935623_{74}-a$$ is a multiple of 15. What is a?

The  $$235935623_{74}-a$$

means 235935623 base 74 minus a.

Jan 28, 2019

#1
+6043
+3

$$235935623_{74} \pmod{15} = \\ 2 + 3(14) + 5+9(14) + 3+5(14)+6+2(14)+3 \pmod{15} = \\ 285 \pmod{15} = 0\\ \text{so }a=0$$

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Jan 28, 2019
#2
+23181
+9

Suppose a is an integer such that 0 \le a \le 14, and $$235935623_{74}-a$$ is a multiple of 15. What is a?
The $$235935623_{74}-a$$
means $$235935623$$ base $$74$$ minus $$a$$.

$$\small{ \begin{array}{|rcll|} \hline && \mathbf{235935623_{74}\pmod{15}} \\ &\equiv& 2\cdot 74^8 +3\cdot 74^7 +5\cdot 74^6 +9\cdot 74^5 +3\cdot 74^4 +5\cdot 74^3 +6\cdot 74^2 +2\cdot 74^1 +3 \pmod{15} \\\\ && \boxed{ 74 \equiv -1 \mod 15} \\\\ &\equiv& 2\cdot (-1)^8 +3\cdot (-1)^7 +5\cdot (-1)^6 +9\cdot (-1)^5 +3\cdot (-1)^4 \\ && +5\cdot (-1)^3 +6\cdot (-1)^2 +2\cdot (-1)^1 +3 \pmod{15} \\\\ &\equiv& \not{2}-\not{3}+\not{5}-9 +3 -\not{5} +6 -\not{2} +\not{3} \pmod{15} \\ &\equiv& -9 +3 +6 \pmod{15} \\ &\equiv& 0 \pmod{15} \\\\ && \mathbf{235935623_{74} -a \equiv 0 \pmod{15} } \quad | \quad 235935623_{74}\equiv 0 \pmod{15} \\ && 0 -a \equiv 0 \pmod{15} \\ && 0 \equiv a \pmod{15} \quad | \quad 0 \pmod{15} = 0 \\ && 0 \equiv 0 \pmod{15}\Rightarrow \mathbf{a=0} \\ \hline \end{array} }$$

Jan 28, 2019