find the last three digits of 4^2^(42) Mohammed Alwan


Question and Answers Forum

All Questions Topic List
Number Theory Questions
Previous in All Question Next in All Question
Previous in Number Theory Next in Number Theory
Question Number 191846 by malwan last updated on 01/May/23

$f i n d t h e l a s t t h r e e d i g i t s$ $o f 4^{2^{42}}$ $M o h a m m e d A l w a n$
	Answered by deleteduser1 last updated on 01/May/23

	$4^{2^{42}} = 2^{2^{43}} \equiv x (m o d 1000) \Rightarrow 2^{2^{43} - 3} \equiv \frac{x}{8} (m o d 125)$ $ϕ (125) = 100$ $2^{43} - 3 \equiv y (m o d 100) \Rightarrow 2^{41} \equiv \frac{y + 3}{4} (m o d 25)$ $ϕ (25) = 20 \Rightarrow 2^{41} \equiv {(2^{20})}^{2} \times 2 \equiv 1 \times 2 \equiv \frac{y + 3}{4} (m o d 25)$ $\Rightarrow 8 \equiv y + 3 (m o d 100) \Rightarrow y \equiv 5 (m o d 100)$ $\Rightarrow 2^{43} \equiv 8 (m o d 100)$ $\Rightarrow 2^{2^{43} - 3} \equiv (2^{100 q}) 2^{5} \equiv \frac{x}{8} (m o d 125)$ $\Rightarrow 8 \times 32 \equiv x (m o d 1000) \Rightarrow x \equiv 256 (m o d 1000)$ $\Rightarrow L a s t t h r e e d i g i t s o f 4^{2^{42}} = 256$
	Commented by malwan last updated on 01/May/23

	$t h a n k y o u s o m u c h s i r$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum