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Intern  B
Joined: 17 Sep 2017
Posts: 43
Re: Which of the following numbers is prime?  [#permalink]

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Quote:
Hence, the units digit of 4^66 + 7^66 is 5 (6+9).

Tks Bunuel. I understood the cycle power part already but I still don't understand how do we have number 5 here? I mean 4^66 + 7^66 then unit digit of it will be 6+9 = 15 but why it turned out to be 5 (6+9)
Retired Moderator D
Joined: 25 Feb 2013
Posts: 1156
Location: India
GPA: 3.82
Re: Which of the following numbers is prime?  [#permalink]

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lichting wrote:
Quote:
Hence, the units digit of 4^66 + 7^66 is 5 (6+9).

Tks Bunuel. I understood the cycle power part already but I still don't understand how do we have number 5 here? I mean 4^66 + 7^66 then unit digit of it will be 6+9 = 15 but why it turned out to be 5 (6+9)

hi lichting

Can you clarify what you were not able to understand? 6+9 is 15 so unit's digit will be 5 and 1 will be carry forward. so unit's digit of 4^66+7^66 will be 5
Intern  B
Joined: 17 Sep 2017
Posts: 43
Re: Which of the following numbers is prime?  [#permalink]

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niks18 wrote:
lichting wrote:
Quote:
Hence, the units digit of 4^66 + 7^66 is 5 (6+9).

Tks Bunuel. I understood the cycle power part already but I still don't understand how do we have number 5 here? I mean 4^66 + 7^66 then unit digit of it will be 6+9 = 15 but why it turned out to be 5 (6+9)

hi lichting

Can you clarify what you were not able to understand? 6+9 is 15 so unit's digit will be 5 and 1 will be carry forward. so unit's digit of 4^66+7^66 will be 5

I'm so sorry T_T
I thought 5 (6+9) is 5 * (6+9) = 5 * 15.
My mistake. Thank you
Intern  B
Joined: 01 Aug 2017
Posts: 12
GMAT 1: 680 Q42 V41
GPA: 4
Which of the following numbers is prime?  [#permalink]

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Bunuel wrote:
lichting wrote:
Quote:
B. 231+331231+331 --> the units digit of 2^31 is 8 and the units digit of 3^31 is 7 (the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}). Hence, the units digit of 2^31 + 3^31 is 5 (8+7). Thus 2^31 + 3^31 is divisible by 5. Not a prime.

C. 466+766466+766 --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6}) and the units digit of 7^66 is 9 (the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}). Hence, the units digit of 4^66 + 7^66 is 5 (6+9). Thus 4^66 + 7^66 is divisible by 5. Not a prime.

I don't understand how do we get 5 here. Is there any formula that I don't know? Need help

$$4^{66} + 7^{66}$$ --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6}) and the units digit of 7^66 is 9 (the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}). Hence, the units digit of 4^66 + 7^66 is 5 (6+9). Thus 4^66 + 7^66 is divisible by 5. Not a prime.

The units digit of 4^(positive integer) repeats in blocks of two - {4, 6}:
4^1 = 4;
4^2 = 16;
4^3 = 8\frac{4[}{fraction];
4^4 = 256;
...

So, the inits digit of 4^256 will be 6 (the odd powers give 4 and even powers give 6).

The units digit of 7^(positive integer) repeats in blocks of four - {7, 9, 3, 1}:
7^1 = 7;
7^2 = 49;
7^3 = ...[fraction]3};
7^4 = ...1;
7^5 = ...7 (7 again)
...

So, the inits digit of 7^66 will be 9. Divide 66 (power) by 4 (cyclisity), remainder is 2. So, the units digit of 7^66 is the same as that of the units digit of 7^2, which is 9.

Hence, the units digit of 4^66 + 7^66 is 5 (6+9).

Theory is here: https://gmatclub.com/forum/math-number- ... 88376.html

Check Units digits, exponents, remainders problems directory in our Special Questions Directory.

Hope it helps.

but why are we multiplying the two units digits by 5 ? why is it 5(6+9)?
Math Expert V
Joined: 02 Sep 2009
Posts: 60605
Re: Which of the following numbers is prime?  [#permalink]

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1
priyatomar wrote:
Bunuel wrote:
lichting wrote:
$$4^{66} + 7^{66}$$ --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6}) and the units digit of 7^66 is 9 (the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}). Hence, the units digit of 4^66 + 7^66 is 5 (6+9). Thus 4^66 + 7^66 is divisible by 5. Not a prime.

The units digit of 4^(positive integer) repeats in blocks of two - {4, 6}:
4^1 = 4;
4^2 = 16;
4^3 = 8\frac{4[}{fraction];
4^4 = 256;
...

So, the inits digit of 4^256 will be 6 (the odd powers give 4 and even powers give 6).

The units digit of 7^(positive integer) repeats in blocks of four - {7, 9, 3, 1}:
7^1 = 7;
7^2 = 49;
7^3 = ...[fraction]3};
7^4 = ...1;
7^5 = ...7 (7 again)
...

So, the inits digit of 7^66 will be 9. Divide 66 (power) by 4 (cyclisity), remainder is 2. So, the units digit of 7^66 is the same as that of the units digit of 7^2, which is 9.

Hence, the units digit of 4^66 + 7^66 is 5 (6+9).

Theory is here: https://gmatclub.com/forum/math-number- ... 88376.html

Check Units digits, exponents, remainders problems directory in our Special Questions Directory.

Hope it helps.

but why are we multiplying the two units digits by 5 ? why is it 5(6+9)?

We are not multiplying we are adding: 6 + 9. The units digit of a positive number ending with 6 PLUS the units digit of a positive number ending with 9 is a number ending with 5.
_________________ Re: Which of the following numbers is prime?   [#permalink] 09 Dec 2019, 06:02

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