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705-805 (Hard)|   Exponents|   Remainders|      
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RenB
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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 26 Sep 2023, 17:42
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E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

85% (hard)

Question Stats:

47% (01:52) correct 53% (02:16) wrong based on 430 sessions

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What is the remainder when 23^44*37^24 + 59^42 - 10 is divided by 6?

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E. 4
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karanpatira
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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 09 Dec 2023, 05:46
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Can,Will
KarishmaB can you give a detailed explanation for this problem
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Answer:

Remainder of k(x)+ c when divided by x is c.
Similarly remainder of k(x) - c when divided by x is -c.

Hence for x=6,

23^44 is similar to (-1)^44 = 1
[23 = (6k-1)]

37^24 is similar to (1)^24=1

59^42 is similar to (-1)^24 =1

Now, rewriting the question, (1*1 + 1 -10)/6 = -8

As remainder is in negative, keep adding 6 till positive remainder.

Answer = -8+12 = 4

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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 26 Sep 2023, 21:28
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(3*7)^24* 3^20+ 59^24-10
21^4+ (60-1)^42-10
Remainder when divided by 6 is then-5+5 -4

Remainder is -4 or 2. How is the ans 4
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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 27 Sep 2023, 08:39
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RenB
What is the remainder when 23^44*37^24 + 59^42 - 10 is divided by 6?

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\(\frac{23}{6} =\) Rem \(5\) & \(\frac{23^2}{6} =\) Rem \(1\)

\(\frac{37}{6} =\) Rem \(1\)

\(\frac{59}{6} =\) Rem \(5\) & \(\frac{59^2}{6} =\) Rem \(1\)

\(\frac{10}{6} =\) Rem \(4\)

Now, \(\frac{23^{44}*37^{24} + 59^{42} - 10}{6}\)

\(= \frac{23^{2*22}}{6}*\frac{37^{24}}{6} + \frac{59^{21*2}}{6} - \frac{10}{6 }\)

\(\frac{= 1*1 + 1 - 4}{6} =>\) Rem \(6 - 2 = 4\), Answer will be (E)
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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 09 Dec 2023, 03:36
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KarishmaB can you give a detailed explanation for this problem
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Can,Will
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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 09 Dec 2023, 20:46
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gmatophobia

still not able to understand the right approach to solve this problem
kindly provide the detailed explanation
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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 09 Dec 2023, 22:39
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Can,Will
gmatophobia

still not able to understand the right approach to solve this problem
kindly provide the detailed explanation
Show more

RenB
What is the remainder when 23^44*37^24 + 59^42 - 10 is divided by 6?

A. 0
B. 1
C. 2
D. 3
E. 4
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Can,Will - Let's take this part by part

  • Remainder(\(\frac{23^{44}}{6}\)) ⇒

    Remainder(\(\frac{23}{6}\)) = -1

    Remainder(\(\frac{23^{44}}{6}\)) = Remainder(\(\frac{(-1)^{44}}{6}\)) = 1

  • Remainder(\(\frac{37^{24}}{6}\)) ⇒

    Remainder(\(\frac{37}{6}\)) = 1

    Remainder(\(\frac{37^{24}}{6}\)) = Remainder(\(\frac{(1)^{24}}{6}\)) = 1

  • Remainder(\(\frac{59^{42}}{6}\)) ⇒

    Remainder(\(\frac{59}{6}\)) = -1

    Remainder(\(\frac{(59)^{42}}{6}\)) = Remainder(\(\frac{(-1)^{44}}{6}\)) = 1

  • Remainder(\(\frac{10}{6}\)) ⇒

    Remainder(\(\frac{10}{6}\)) = 4

Remainder (\(\frac{23^{44}*37^{24} + 59^{42} - 10}{6}\)) ⇒ Remainder (\(\frac{1 * 1 + 1 - 4}{6}\)) = Remainder (\(\frac{2 -4}{6}\)) = \(-2 \)

As remainders cannot be negative, actual remainder = \(6 - 2 = 4\)

Option E
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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 09 Dec 2023, 22:51
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Identify the themes in the units place.
3 has a recurring theme of units digit being 3,9,7,1,3,9,7,1…

For 7 it is 7,9,3,1,7,9,3,1…

For 9 it is 9,1,9,1

Find the units digit for each number based off the power it is raised to. For this question it is 1, 1, & 9

1*1 + 9 =10
Therefore the remainder is 4 or -2

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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 09 Dec 2023, 23:29
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RenB
What is the remainder when 23^44*37^24 + 59^42 - 10 is divided by 6?

A. 0
B. 1
C. 2
D. 3
E. 4
Show more

First check this post:
https://anaprep.com/number-properties-m ... emainders/

\(23^{44} *37^{24} + 59^{42} - 10 = (24 - 1)^{44}*(36 + 1)^{24} + (60 - 1)^{42} - 10\)

Now we need to use Binomial theorem here to say that every term of
\((24 - 1)^{44}*(36 + 1)^{24}\) is divisible by 6 except the last one which is 1.

Also every term of \((60 - 1)^{42} \) is divisible by 6 except the last one which is 1.

So I will get

\((23^{44} *37^{24}) + (59^{42}) - 10 = (6a + 1) + (6b + 1) - 10 = 6a + 6b - 8 = 6a + 6b - 6 - 2\)
Hence remainder here will be -2 which is a positive remainder of 4.

Answer (E)

You can try another question on negative remainders here: https://anaprep.com/number-properties-t ... emainders/
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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 17 Jan 2025, 21:34
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gmatophobia KarishmaB can you please provide link to this negative remainder concept?

Thanks!


gmatophobia
Can,Will
gmatophobia

still not able to understand the right approach to solve this problem
kindly provide the detailed explanation

RenB
What is the remainder when 23^44*37^24 + 59^42 - 10 is divided by 6?

A. 0
B. 1
C. 2
D. 3
E. 4

Can,Will - Let's take this part by part

  • Remainder(\(\frac{23^{44}}{6}\)) ⇒

    Remainder(\(\frac{23}{6}\)) = -1

    Remainder(\(\frac{23^{44}}{6}\)) = Remainder(\(\frac{(-1)^{44}}{6}\)) = 1

  • Remainder(\(\frac{37^{24}}{6}\)) ⇒

    Remainder(\(\frac{37}{6}\)) = 1

    Remainder(\(\frac{37^{24}}{6}\)) = Remainder(\(\frac{(1)^{24}}{6}\)) = 1

  • Remainder(\(\frac{59^{42}}{6}\)) ⇒

    Remainder(\(\frac{59}{6}\)) = -1

    Remainder(\(\frac{(59)^{42}}{6}\)) = Remainder(\(\frac{(-1)^{44}}{6}\)) = 1

  • Remainder(\(\frac{10}{6}\)) ⇒

    Remainder(\(\frac{10}{6}\)) = 4

Remainder (\(\frac{23^{44}*37^{24} + 59^{42} - 10}{6}\)) ⇒ Remainder (\(\frac{1 * 1 + 1 - 4}{6}\)) = Remainder (\(\frac{2 -4}{6}\)) = \(-2 \)

As remainders cannot be negative, actual remainder = \(6 - 2 = 4\)

Option E
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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 18 Jan 2025, 02:12
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Please find below some links to useful resources on Remainders:

All About Negative Remainders on the GMAT

Tips and Hints on Remainders

Theory on Remainders

More Tips and Tricks for Remainder problems

Hope it helps!


Sujithz001
gmatophobia KarishmaB can you please provide link to this negative remainder concept?

Thanks!


gmatophobia
Can,Will
gmatophobia

still not able to understand the right approach to solve this problem
kindly provide the detailed explanation

RenB
What is the remainder when 23^44*37^24 + 59^42 - 10 is divided by 6?

A. 0
B. 1
C. 2
D. 3
E. 4

Can,Will - Let's take this part by part

  • Remainder(\(\frac{23^{44}}{6}\)) ⇒

    Remainder(\(\frac{23}{6}\)) = -1

    Remainder(\(\frac{23^{44}}{6}\)) = Remainder(\(\frac{(-1)^{44}}{6}\)) = 1

  • Remainder(\(\frac{37^{24}}{6}\)) ⇒

    Remainder(\(\frac{37}{6}\)) = 1

    Remainder(\(\frac{37^{24}}{6}\)) = Remainder(\(\frac{(1)^{24}}{6}\)) = 1

  • Remainder(\(\frac{59^{42}}{6}\)) ⇒

    Remainder(\(\frac{59}{6}\)) = -1

    Remainder(\(\frac{(59)^{42}}{6}\)) = Remainder(\(\frac{(-1)^{44}}{6}\)) = 1

  • Remainder(\(\frac{10}{6}\)) ⇒

    Remainder(\(\frac{10}{6}\)) = 4

Remainder (\(\frac{23^{44}*37^{24} + 59^{42} - 10}{6}\)) ⇒ Remainder (\(\frac{1 * 1 + 1 - 4}{6}\)) = Remainder (\(\frac{2 -4}{6}\)) = \(-2 \)

As remainders cannot be negative, actual remainder = \(6 - 2 = 4\)

Option E
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What is the remainder when 23^44 x 37^24 + 59^42-10 is divided by 6? 18 Jan 2025, 02:17
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Here is my blog post on negative remainders:
https://gmatclub.com/forum/all-about-ne ... seI6Aj48jO
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