What is the number of digits of the following product?
11 * 11011 * 1111 * 1010 * 11100111
(In reply to
Puzzle Answer by K Sengupta)
Let the given expression be N (say).then,
N= 11x11011x1111x1010x11100111
= 11 * (11*1001) * (11*101) * (101*10) * (111*10001)
= 11^3 * 10 * 101^2 * 1001 * 111 * 100001
= 1331*10*10201*1001*111*100001
= 1.331 * 10^3 * 10* 1.0201 * 10^4 *1.001 * 10^3 *1.11 *10^2 * 1.00001 *10^5
= (1.331 * 1.0201 * 1.001 * 1.11 * 1.00001) * 10^18
= M * 10^18, where:
M = 1.331 *1.0201 * 1.001 * 1.11*1.00001 .....(#)
Now,
1.331 < 1.4
1.0201 < 1.1
1.11< 1.2
Thus, we must have:
M
< 1.4 * 1.1 * 1.001 * 1.2 * 1.00001
= (1.4* 1.1*1.2) * 1.001 * 1.00001
= 1.848 * 1.001* 1.00001
= 1.849848 * 1.00001
= 1.849818498
< 1.85
Since each of the multiplicand in M is >1, it follows that M must be > 1
Accordingly, M (- (1, 1.85)
Therefore, N = M* 10^18, where 1< M < 1.85
Consequently, the given expression when evaluated, must contain precisely 19 digits.
**** I solved this puzzle entirely by hand, using the calculator only for verification.
Edited on June 19, 2022, 4:59 am
Explanation to the Puzzle Answer
