If 4^10 - 1 = 1048a75 then find the value of "a" using divisibility rules.

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Hello Everyone. Before going to solve the question, let us go and understand divisibility rules of 2, 4, 8,…..  as it is an essential thing that should be known for understanding this approach of finding solution.

Now, let us go to the divisibility rules of 2, 4, 8,......

Divisibility rule of 2 :

A number is divisible by 2 if its units place digit is one of the digits 0, 2, 4, 6, 8.

Divisibility rule of 4 :

A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

Divisibility rule of 8 :

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

Similarly, the divisibility rules of 16, 32, 64, .... follow the same pattern as we can see in the above divisibility rules.

Now, we have understood the divisibility rules of 2, 4, 8, 16, 32,........

So, now let us go into the question.

What to find : 

Value of  "a" that satisfies 4^10 - 1 = 1048a75, by using only divisibility rules without actually calculating 4^10 - 1. 

Solution :

Given that 4^10 - 1 = 1048a75,

Then, 4^10 = 1048a75 + 1 = 1048a76

If we observe the left side of the above equation, it is 4^10. Then we can write like 4^10 = 2^20. This means that the left hand side of the equation is divisible by 1, 2, 4, 8, 16, 32, ...., 2^20. Then, 1048a76 is also divisible by 1, 2, 4, 8, 16, 32, ...., 2^20 as 4^10 = 1048a76.

Now, let us apply divisibility rules to find the value of a.

As the last digit(Unit's place digit) is 6, 1048a76 is divisible by 2. 

As the number formed by the last two digits i.e. 76 is divisible by 4, 1048a76 is divisible by 4.

We cannot find value of "a" using the above two divisibility rules of 2 and 4 respectively as they are already getting satisfied.

We know that 1048a76 is divisible by 8. Therefore, a76 should be divisible by 8 as per the divisibility rule of 8. 

Consider (a76/8) = (100*a+76)/8 = 12*a + 9 +((a + 1)/2) 

We know, that 1048a76 is divisible by 8, therefore, a76 should be divisible by 8.

For that, (a + 1) should be divisible by 2. From this, we can say that "a" can be an odd digit. 

Therefore, "a" can be 1,3,5,7, or 9 only as a digit can vary from 0 to 9 only.

We know that 1048a76 is divisible by 16. Therefore, 8a76 should be divisible by 16 as per the divisibility rule of 16.

Consider (8a76)/16 = (8000 + 100*a + 76)/16 = 6*a + 4 + ((a + 3)/4)

We know that 1048a76 is divisible by 16, therefore, 8a76 should be divisible by 16.

For that, (a + 3) should be divisible by 4, where "a" can be 1, 3, 5, 7, 9.

From this, we can say "a" can be 1, 5, or 9 only.

We know that 1048a76 is divisible by 32. Therefore, 48a76 should be divisible by 32 as per the divisibility rule of 32.

Consider (48a76)/32 = (48000 + 100*a + 76)/32 = 3*a + 1502 + ((a + 3)/8)

We know that 1048a76 is divisible by 32, therefore, 48a76 should be divisible by 32.

For that (a + 3) should be divisible by 8, where "a" can be 1, 5, 9. From this, we can say "a" should be equal to 5.

Therefore, value of "a" satisfying the given equation is 5.

Question to think : Why the divisibility rule of 4 mentioned above is valid ?


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