+537 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.
+6251 \(235935623_{74} \pmod{15} = \\ 2 + 3(14) + 5+9(14) + 3+5(14)+6+2(14)+3 \pmod{15} = \\ 285 \pmod{15} = 0\\ \text{so }a=0 \)
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+26393 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 \(a\).
\(\small{ \begin{array}{|rcll|} \hline && \mathbf{235935623_{74}\pmod{15}} \\ &\equiv& 2\cdot 74^8 +3\cdot 74^7 +5\cdot 74^6 +9\cdot 74^5 +3\cdot 74^4 +5\cdot 74^3 +6\cdot 74^2 +2\cdot 74^1 +3 \pmod{15} \\\\ && \boxed{ 74 \equiv -1 \mod 15} \\\\ &\equiv& 2\cdot (-1)^8 +3\cdot (-1)^7 +5\cdot (-1)^6 +9\cdot (-1)^5 +3\cdot (-1)^4 \\ && +5\cdot (-1)^3 +6\cdot (-1)^2 +2\cdot (-1)^1 +3 \pmod{15} \\\\ &\equiv& \not{2}-\not{3}+\not{5}-9 +3 -\not{5} +6 -\not{2} +\not{3} \pmod{15} \\ &\equiv& -9 +3 +6 \pmod{15} \\ &\equiv& 0 \pmod{15} \\\\ && \mathbf{235935623_{74} -a \equiv 0 \pmod{15} } \quad | \quad 235935623_{74}\equiv 0 \pmod{15} \\ && 0 -a \equiv 0 \pmod{15} \\ && 0 \equiv a \pmod{15} \quad | \quad 0 \pmod{15} = 0 \\ && 0 \equiv 0 \pmod{15}\Rightarrow \mathbf{a=0} \\ \hline \end{array} }\)
