If a = 222+1 and b = 525+1, how many digits does a·b have?

Question

If  a = 2^{22}+1  and  b = 5^{25}+1 , how many digits does  a·b  have?

A.  22 
 
B.  25 
 
C.  27 
 
D.  47 
 
E.  49

chetan2u · Accepted Answer

As  1  is very a small amount as compared to  2^{22}  and  5^{25} , we can take 
 a =( 2^{22}+1 ) ~  2^{22}  and  b = (5^{25}+1 )~ 5^{25}

a·b =  2^{22}*  5^{25}=(2*5)^{22}*5^3=10^{22}*125 
 10^{22}  means twenty two of 0 digit and 125 in front of these 22 zeroes, so 3+22=25 digits

Enchanting · Answer

a.b = (2^{22} + 1)(5^{25} + 1)=2^{22}.5^{25} + 2^{22} + 5^{25} + 1 = 10^{22}*5^3 + 2^{22} + 5^{25} + 1=125*10^{22} + 2^{22} + 5^{25} + 1 
  The number of digits of a.b = number of digits of  125*10^{22} 
  Let’s take a small example for better understanding.
 Let  c = 2^2+1  and  d =5^3+1, c.d = 2^2*5^3 + 2^2 + 5^3 + 1 = 5*10^2 + 4 + 125 + 1 = 500 + 130 = 630 
 The number of digit of c.d = The number of digit of  2^2*5^3 
  The number of digits of  125*10^{22} = 3 + 23 – 1 = 25    
 ANSWER : - B

yashikaaggarwal · Answer

A = 2^22+1
B = 5^25+1
Let A = 2^22 (approximate)
Let B = 5^25 (approx)

Log A = Log2^22
Log B = Log5^25
Log X^Y = Y*LogX
Therefore
Log A = 22*Log2
Log B = 25*Log5
Log2 = 0.3 and Log5 = 0.7
A = 22*0.3 = 6.6 or 7 digit at max.
B = 25*0.7 = 17.5 or 18 digit at max.
10^18*10^7 = 10^25 digits

Answer is B

Posted from my mobile device

MathRevolution · Answer

=>

a·b = (2^{22} + 1)(5^{25} + 1) = 2^{22}·5^{25} + 5^{25} + 2^{22} + 1 = 5^3·(2·5)^{22} + 5^{25} + 2^{22} + 1
= 125·10^{22} + 5^{25} + 2^{22} + 1.

Therefore,  a·b  has  25  digits, as that is the largest exponent.

Therefore, B is the answer.
Answer: B

angelthomas · Answer

is the number of digits of a number determined by its largest exponent?

chetan2u · Answer

No, that will not be correct.
The solution above your post is wrong because the answer 25 does not come from  5^{25}  but from  125*10^{22} .
For example  5^3=125  and  5^4=625 . As you can see that powers are 3 and 4, but digits are 3 in each case.

Thus, the answer will depend on the power as well as the base. If base is 10, then ONLY you can be 100% sure of number of digits as (power+1)

MathRevolution · Answer

As we are dealing here with the number of digits for the product of a*b, yes the highest power of exponents will give us the number of digits.

But we need to simplify first and combine the common powers first and then we check the highest power.

Thanks

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If a = 222+1 and b = 525+1, how many digits does a·b have?

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If a = 222+1 and b = 525+1, how many digits does a·b have?

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