How many ordered triples of positive integers (a,b,c) satisfy

Question

How many ordered triples of positive integers (a,b,c) satisfy  (a^{b})^{c}=4  ?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Bunuel · Answer

4 = 2^2 = 4^1

(2^{2})^{1}=4

(2^{1})^{2}=4

(4^{1})^{1}=4

Answer: C.

P.S. If we were not restricted to POSITIVE integers only, then there would be two more triplets:
 ((-2)^{2})^{1}=4

((-2)^{1})^{2}=4

Vsolo · Answer

Why can't the answer be 4 ordered pairs? (a,b,c) : (4,1,1) and (a,c,b) : (4,1,1)?

egmat · Answer

Vsolo

I can see why you're confused - this is actually a common misconception about ordered triples! Let me clarify what's happening here.

The Key Misunderstanding

An ordered triple  (a,b,c)  is a  single mathematical object  consisting of three numbers in a specific order. When we write   (4,1,1)  , that's ONE ordered triple, not two. The notation  (a,b,c)  simply means:

a  = first position
 b  = second position 
 c  = third position

These are just  labels for positions , not variables we can rearrange!

Why Your Approach Doesn't Work

When you write  (a,b,c): (4,1,1)  and  (a,c,b): (4,1,1) , you're not creating two different ordered triples. You're just relabeling the same triple  (4,1,)  in a confusing way.

Think of it this way: If I have a phone number (555-1234), I can't count it as two different phone numbers by calling the digits different names!

The Correct Solution

For   a^{bc} = 4 , we need to find ALL distinct ordered triples:

Case 1:  If  a = 4 
 4^{bc} = 4  means  bc = 1  
Only solution:  b = 1, c = 1

Triple:   (4,1,1)

Case 2:  If  a = 2

2^{bc} = 4 = 2^2  means  bc = 2

Solutions:  b = 1, c = 2  OR  b = 2, c = 1

Triples:   (4,2,1)   and   (4,1,2)

Total: 3 ordered triples

If you'd like, you can practice similar quant questions  here - look for questions involving counting methods, ordered pairs/triples, and exponent properties to reinforce when order creates distinct solutions versus when it doesn't.

Key Takeaway:

Each ordered triple is unique based on  which number appears in which position . The triple   (4,1,1)   means  a=4, b=1, c=1  specifically, and that's just one solution, regardless of how we label the positions.

Remember:  In ordered pairs/triples, order matters -  (4,2,1)  and  (4,1,2)  are different because the numbers appear in different positions, but  (4,1,1)  is always just  (4,1,1) !

athakar7 · Answer

what is significance of ordered here

egmat · Answer

athakar7  This is actually a crucial distinction that affects our final answer!

Understanding "Ordered" vs "Unordered"

When we say  ordered triples , it means the  position matters . The triple  (a, b, c)  represents:
-  a  is the  first  element (the base)
-  b  is the  second  element (the first exponent) 
-  c  is the  third  element (the second exponent)

Why This Matters in Our Problem:

The expression  (a^b)^c  depends on which number goes where. For example:
-  (2, 1, 2)  means  (2^1)^2 = 2^2 = 4  ✓
-  (2, 2, 1)  means  (2^2)^1 = 4^1 = 4  ✓

Even though both use the same three numbers ( 1, 2, 2 ), they're  different ordered triples  because the numbers appear in different positions.

If the problem asked for "unordered" sets: 
We would count  \{1, 2, 2\}  only once, regardless of arrangement. But since it asks for  ordered  triples, we count  (2, 1, 2)  and  (2, 2, 1)  as two distinct solutions.

How many ordered triples of positive integers (a,b,c) satisfy

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How many ordered triples of positive integers (a,b,c) satisfy

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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