1320.11 – The Number of Digits in a Product

Let PP be 3,659,893,456,789,325,678×342,973,489,379,2563,659,893,456,789,325,678 \times 342,973,489,379,256.

The number of digits in PP is:

  1. 36

  2. 35

  3. 34

  4. 33

  5. 32

Solution

Rounding to the nearest godzillionth, the number of digits in PP will be the same as the number of digits in 3.7×10193.7 \times 10^{19} times 3.4×10143.4 \times 10^{14}. (This is called scientific notation.) Thus we seek the number of digits in

P≈(3.7×1019)×(3.4×1014)=12.58×1019+14=1.258×1033+1=1.258×1034\begin{aligned}P &\approx (3.7 \times 10^{19}) \times (3.4 \times 10^{14}) \\&= 12.58 \times 10^{19 + 14} \\&= 1.258 \times 10^{33 + 1} \\&= 1.258 \times 10^{34}\end{aligned}

The number of digits is 34 and the answer is (c).