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Let a,b,c be positive integers less than 10 such that (100a + 10b + c)² = (a + b + c)⁵, what is the value of (a × b - c)?
[Answer : a × b - c = 5, a = 2, b = 4, and c = 3]
Answers
Step-by-step explanation:
Given :-
Let a,b,c be positive integers less than 10 such that (100a + 10b + c)² = (a + b + c)⁵.
To find :-
What is the value of (a × b - c)?
Solution :-
Given that
a,b,c are three positive integers and less than 10
So, possible values of a ,b,c are any of the three digits among 1,2,3,4,5,6,7,8,9.
Given condition is
(100a + 10b + c)² = (a + b + c)⁵ ----------(1)
On taking square root both sides then
=> [(100a + 10b + c)²]^½ = [(a + b + c)⁵]^½
=> (100a + 10b + c) = (a + b + c)^5/2
Since (a^m)^n = a^mn
=> abc = (a + b + c)^5/2
Where abc is the three digits number
abc is equal to the 5/2th power of a+b+c.
So (a+b+c) must be an integer lying between 3 and 27 .
3 ≤ (a+b+c)≤ 27
(a+b+c) is a perfect square and whose 5/2 th power is a three digit number.
The perfect square numbers between 3 and 27 are 4,9,16 and 25
Checking for 4 :-
If a+b+c = 4 then
(a+b+c)^5/2 = (4)^5/2
=>(2^2)^5/2
=> 2^(2×5/2)
=>2^5
=> 2×2×2×2×2
=> 32
But it is not a three digit number.
Checking for 9:-
If a+b+c = 9 then
(a+b+c)^5/2 = (9)^5/2
=>(3^2)^5/2
=> 3^(2×5/2)
=>3^5
=> 3×3×3×3×3
=> 243
it is a three digit number.
Checking for 16:-
If a+b+c = 4 then
(a+b+c)^5/2 = (16)^5/2
=>(2^4)^5/2
=> 2^(4×5/2)
=>2^(2×5}
=> 2^10
=> 2×2×2×2×2×2×2×2×2×2
=> 1024
But it is not a three digit number.
Checking for 25:-
If a+b+c = 4 then
(a+b+c)^5/2 = (25)^5/2
=>(5^2)^5/2
=> 5^(2×5/2)
=>5^5
=> 5×5×5×5×5
=> 3125
But it is not a three digit number.
among 32,243,1024,3125 the three digit number is 243 and
The sum of the digits =2+4+3= 9
So among all square we get only three digit number for (a+b+c) = 9
So, 243 is the only number that satisfies the given conditions.
So we get abc = 243
So, (100a+10b+c) = 243
Now,
From (1)
(100a + 10b + c)² = (a + b + c)⁵
=> (243)² =9⁵
=> (2+4+3)⁵
We have ,
a = 2
b = 4
c = 3
Now,
The value of (a×b-c)
=> (2×4-3)
=>8-3
=> 5
Therefore, (a×b-c) = 5
Answer:-
The value of a = 2
The value of b = 4
The value of c = 3
The value of a×b-c = 5
Used formulae:-
- (a^m)^n = a^mn
Step-by-step explanation:
Step-by-step explanation:
Given :-
Let a,b,c be positive integers less than 10 such that (100a + 10b + c)² = (a + b + c)⁵.
To find :-
What is the value of (a × b - c)?
Solution :-
Given that
a,b,c are three positive integers and less than 10
So, possible values of a ,b,c are any of the three digits among 1,2,3,4,5,6,7,8,9.
Given condition is
(100a + 10b + c)² = (a + b + c)⁵ ----------(1)
On taking square root both sides then
=> [(100a + 10b + c)²]^½ = [(a + b + c)⁵]^½
=> (100a + 10b + c) = (a + b + c)^5/2
Since (a^m)^n = a^mn
=> abc = (a + b + c)^5/2
Where abc is the three digits number
abc is equal to the 5/2th power of a+b+c.
So (a+b+c) must be an integer lying between 3 and 27 .
3 ≤ (a+b+c)≤ 27
(a+b+c) is a perfect square and whose 5/2 th power is a three digit number.
The perfect square numbers between 3 and 27 are 4,9,16 and 25
Checking for 4 :-
If a+b+c = 4 then
(a+b+c)^5/2 = (4)^5/2
=>(2^2)^5/2
=> 2^(2×5/2)
=>2^5
=> 2×2×2×2×2
=> 32
But it is not a three digit number.
Checking for 9:-
If a+b+c = 9 then
(a+b+c)^5/2 = (9)^5/2
=>(3^2)^5/2
=> 3^(2×5/2)
=>3^5
=> 3×3×3×3×3
=> 243
it is a three digit number.
Checking for 16:-
If a+b+c = 4 then
(a+b+c)^5/2 = (16)^5/2
=>(2^4)^5/2
=> 2^(4×5/2)
=>2^(2×5}
=> 2^10
=> 2×2×2×2×2×2×2×2×2×2
=> 1024
But it is not a three digit number.
Checking for 25:-
If a+b+c = 4 then
(a+b+c)^5/2 = (25)^5/2
=>(5^2)^5/2
=> 5^(2×5/2)
=>5^5
=> 5×5×5×5×5
=> 3125
But it is not a three digit number.
among 32,243,1024,3125 the three digit number is 243 and
The sum of the digits =2+4+3= 9
So among all square we get only three digit number for (a+b+c) = 9
So, 243 is the only number that satisfies the given conditions.
So we get abc = 243
So, (100a+10b+c) = 243
Now,
From (1)
(100a + 10b + c)² = (a + b + c)⁵
=> (243)² =9⁵
=> (2+4+3)⁵
We have ,
a = 2
b = 4
c = 3
Now,
The value of (a×b-c)
=> (2×4-3)
=>8-3
=> 5
Therefore, (a×b-c) = 5
Answer:-
The value of a = 2
The value of b = 4
The value of c = 3
The value of a×b-c = 5
Used formulae:-
(a^m)^n = a^mn