If b is a positive integer and b^4 is divisible by 81

Question

If b is a positive integer and b^4 is divisible by 81 , which of the following could be the remainder when b is divided by 9? A. 1 B. 2 C. 4 D. 6 E. 8 c99069b3-d78d-4263-bd9b-6c3f36080a97.jpg

gmatophobia · Accepted Answer

Providing some more context to VivekPrateek's explanation -

b^4  is divisible by 81 indicating that the prime factorized form of  b  contains a  3 .

Therefore

b = 3 * x  ⇒ x is some other factor

Remainder ( \frac{b}{9} ) = Remainder ( \frac{3x }{ 9} ) = Remainder ( \frac{x}{3} )

When a number is divided by 3, the possible remainder is either 0, 1, or 2.

However as we had canceled 3 from the numerator and denominator, the remainder when 3x is divided by 9 is (0*3), (1*3), or (2*3).

The possible remainders can be 0, 3, or 6.

Hence    Option D

VivekSri · Answer

I will go with option D

b^4 is divisible by 81, it means b is multiple by 3. Since b is multiple of 3,So when you divide b with 9, the valid remainder could be 0,3,6 and the possible answer is 6

GMAT700SSE · Answer

My approach was:

2 Equations
b^4/81 = integer
b/9 = integer + reminder

than i factored the divisor
b^4/3^4 = b/3 = integer
and
b/3^2 = interger + reminder

than i started testing values for b starting with 3 as b/3 has to be integer

3/3 = 1
3/9 = 0 + 6 as a reminder

=> Option D

sayan640 · Answer

Can you please explain the highlighted part of your explanation ?  gmatphobia   Bunuel  ..dint understand completely..

Aim4gmat2024 · Answer

(B^4)/(3^4). it implies B is multiple of 3 . So reminder is 3 or 6 and so on. From the options it is 6

LucasH20 · Answer

Hey  gmatophobia  &  Bunuel ,

could you please tell me, if my alternative approach would also work to solve questions like this?

We know that b^4=81*k, thus b=3*(fourth root of k). Now we can use a value for k such that we receive an integer for b, starting with the first being k=1, the remainder would be (3*1)/9 = 3, not a possible answer choice. Going to the next choice being k=16, the remainder is (3*2)/9=6, possible answer choice, thus answer is remainder could be 6.

Kind regards
Lucas

Bunuel · Answer

Yes, that's correct. +1.

DanTheGMATMan · Answer

https://www.youtube.com/watch?v=kY3DCYnvJrc

egmat · Answer

Hey there! I see you're working on this divisibility problem – these can be tricky because they require you to think about the relationship between  b  and  b^4 . Let me walk you through how to approach this systematically.

Here's the key insight you need:

When we're told that  b^4  is divisible by 81, let's first think about what 81 really is. Notice that  81 = 3^4 . This is crucial because it tells us something important about  b  itself.

Step 1: Understanding the constraint on b

Think about this: if  b^4  must be divisible by  3^4 , then  b^4  needs to have at least four factors of 3.

Now, if  b  has  n  factors of 3, then  b^4  will have  4n  factors of 3 (because we're multiplying  b  by itself 4 times). For  b^4  to be divisible by  3^4 , we need:

4n \geq 4

This means  n \geq 1

So  b  must have at least one factor of 3 – in other words,  b  must be divisible by 3!

Step 2: Finding possible remainders

Since  b  must be a multiple of 3, let's check what remainders are possible when we divide multiples of 3 by 9:

-  3 \div 9  gives remainder 3
-  6 \div 9  gives remainder 6 
-  9 \div 9  gives remainder 0
-  12 \div 9  gives remainder 3
-  15 \div 9  gives remainder 6
-  18 \div 9  gives remainder 0

Notice the pattern? Multiples of 3, when divided by 9, can only have remainders of 0, 3, or 6.

Step 3: Check the answer choices

Looking at our options (1, 2, 4, 6, 8), only  6  is among the possible remainders we found.

Let's verify with  b = 6 : We get  b^4 = 1296 = 81 	imes 16  ✓

Answer: D (6)

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You can check out the  step-by-step solution on Neuron by e-GMAT  to master the systematic framework for all divisibility problems involving powers. The full solution shows you alternative approaches and a powerful pattern that applies to similar GMAT questions. You can also explore other GMAT official questions with detailed solutions on Neuron for structured practice  here .

If b is a positive integer and b^4 is divisible by 81

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