1320.11 – The Number of Digits in a Product

Let PP be 3,659,893,456,789,325,678 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).