 # o mundo dos números

With gadgets becoming a part of us,
an exercise (here) was planned at a
busy avenue in a café named Das Brot.

The questions are the following.
With no rewards to those who answer
rightly or wrongly, one is but
left, right to chance.

Here we go, we begin with 37.
The product of a number with three,
when added sweet little sixteen,
shall produce this very thirty and sevens. Another such  number sports power.
When added before a number,
it is ignored by many.
as an unwanted nuisance.

Add it after a number and people
indeed throng like bees.
So shall we count the digits
in two to the power of mighty seven
multiplied by some ten to the power of five squared ?

Oh my, this multiplication is
must you try.
25 * 25 is 625 and
105 * 105 is 11025.
Spot the pattern and thou shall reap the reward.

Many such questions were devised
with the hope that people shall attempt it.
The responses were a mere five
Wonder I, tremulously, why, why, why?

Gosh,
what need does math have
with such an easy punch of a calculator,
than ever before?

Tidbits

• 37 = 16 + 21 = 16 + 3 * 7
37 is also a prime number; it can be written as 4^3-3^3; 5^3-4^3-3^3+3; and 1 + 6^2; you can try more combinations​. ​An excellent property of 37 can be viewed here.
• 2^7*10^25 = 2 ^2* 2^2 * 2^2 *2 * 10^25 = 128 followed by 25 zeroes; hence the final number has 28 digits​. This question can be posed also as 2^7 * (5*2)^25; expanding the brackets and adding the exponents we would get (2^32 * 5 ^25); For comparison, Avagadro’s number has 24 digits. ​
• 1/4 – 1/8 = 25 % – 12.5% = 12.5 % = 1/8; This can be extended to x^n – x ^(n+1) and you will find that for any integer, the difference comes to (x-1)/x^(n+1);
• 95 * 95 = 902​5​; (10a+5)(10a+5)=(100a^*2+100a+25 = 100a(a+1) + 25;​ In our example, 9 is multiplied by 9+1 to get 90 and this becomes the first two digits of this number; the last two digits will always be 25.​
• 10^3.5 = 10^3 * 10^0.5 = 1000 * ~3 something; hence greater than 100 but less than 300, 400, 1000 and not less than 300.