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[tex]\large\underline{\text{Question 1.}}[/tex]
Unit digit cycle of the power of 7
[tex]\implies7\rightarrow9\rightarrow3\rightarrow1\rightarrow\cdots[/tex]
[tex]\text{(Given value)}[/tex]
[tex]=7^{105}[/tex]
[tex]=(7^{4})^{26}\cdot7[/tex]
The unit digit is hence 7.
[tex]\large\underline{\text{Question 2.}}[/tex]
[tex]\text{(Given value)}[/tex]
[tex]=3^{65}\times6^{59}\times7^{71}[/tex]
[tex]=(21)^{65}\times6^{59}\times7^{6}[/tex]
Unit digit cycle of the power of 21
[tex]\implies1\rightarrow1\rightarrow\cdots[/tex]
Unit digit cycle of the power of 6
[tex]\implies6\rightarrow6\rightarrow\cdots[/tex]
Hence, the unit digit is obtained from the following product.
[tex]\text{(Product)}[/tex]
[tex]\equiv1\times6\times7^{6}\mod10[/tex]
[tex]\equiv1\times6\times9\mod10[/tex]
[tex]\equiv4\mod10[/tex]
Step-by-step explanation:
solution :