modular arithmetic

How would one use modular arithmetic to find the units digit of 7^7^7?

tens digit of 2^65 ?

Note by Alan Liang
8 years, 3 months ago

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3 votes

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Comments

FOR TENS DIGIT OF 2^65, FIND ITS MOD 100.

NOTE THAT 2^10=1024 = 24 (MOD 100) sO 2^20 = 576(MOD 100) = -24(MOD100) 2^40= (-24)^2 (MOD 100) = -24 (MOD100) AGAIN! 2^60 = 2^20 *2^40 (MOD 100) = 576(MOD100) = -24(MOD100)

ALSO, 2^5 = 32(MOD100) SO 2^65=2^60 * 2^5(MOD 100) = (-24)(32)(MOD100) = 32(MOD100)

SO THE TENS DIGIT IS 3 AND THE UNITS DIGIT IS 2.

Shourya Pandey - 8 years, 3 months ago

The first problem :

777733(mod10)7^{7^{7}} \equiv 7^{3} \equiv 3 \pmod {10} and hence the unit digit is 33. (The powers of 77 form a cycle mod 1010. )

Zi Song Yeoh - 8 years, 3 months ago

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how can you equate power 7 as power 3. I guess this is wrong. Please clarify.

Namra Aziz - 8 years, 3 months ago

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no..its perfectly alright..

Nishanth Hegde - 8 years, 3 months ago

777374733(mod10)7^{7} \equiv 7^{3}\cdot7^{4} \equiv 7^{3} \equiv 3 \pmod{10}. Since 721(mod10)7472721(mod10)7^2 \equiv -1 \pmod {10} \Rightarrow 7^{4} \equiv 7^{2}\cdot7^{2} \equiv 1 \pmod{10}

Zi Song Yeoh - 8 years, 3 months ago

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@Zi Song Yeoh Just use euler's theorum for 747^4, since phi(10) = 4

Harshit Kapur - 8 years, 3 months ago

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@Harshit Kapur Yes.

Zi Song Yeoh - 8 years, 3 months ago

But unit digit would be 7.

Ram Prakash Patel - 8 years, 3 months ago

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Zi Song is right. The answer is 3.

Rohan Rao - 8 years, 3 months ago
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