Question

What is the highest power of 91 that divides 78! ?
Platinum is sold in bars of weights ranging from 17 grams to 1760 grams in 7 gram increments. Each bar is sealed in an opaque box weighing 47 grams . The box used for packaging has no marks on it indicating the weight of the bar inside.
The precious metals merchant selling the boxes has put the packed bars into shelves based on weight. However , to be certain , he weighs the packed box in an equal arms two pan balance, and a set of weights( which is common for all the bars and which he considers as “standard” weights.). Each of these weighs an integral number of grams, and have their weight marked on them. The merchant, a superstitious man, always places the packed box of platinum on the left pan, and places the appropriate weights in the left pan or the right pan or both until balance is achieved. This suffices hi m to tell the weight of the packed bar.
What is the minimum number of “standard” weights the merchant must have to be able to accurately determine the weight of all his packed boxes?

Answer 1

Answer:

1 gm  , 7 gm , 21 gm , 63 g  , 189gm , 567 gm , 1701 gm

Step-by-step explanation:

Weight range from 17 gm to 1760 gm

Box weight = 47 gm

So weight to be measured from 47 + 17 = 64 gm   to  1807 gm

64 = 57 + 7 = 1 + 7 *9

1807 = 57 + 1750 = 1 + 7*258

All other weight in between

We can take one weight of 1 gm  = 57 grams = 7 * 8  + 1  

then to measure multiples of 7 we can take

other weights multiples of 7  (  1  , 3 , 9 , 27 , 81 , 243)

47 + 17 = 64 = 1 + 7*9

47 + 24 = 71 = 1 + 7*9 + 7*1

47 + 31 =78 =  1  + 7*9 + 7*3 - 7*1

and so on

47 + 1753 = 1800 = 1 + 243*7 + 7*27 - 7*9 - 7*3 - 7*1

47 + 1760 1807 = 1 + 243*7 + 7*27 - 7*9 - 7*3

Weight required = 1 gm  , 7 gm , 21 gm , 63 g  , 189gm , 567 gm , 1701 gm

highest power of 91 that divides 78!

91 = 13 * 7

Let see how many 13 & 7 factor before 78 (inclusing 78)

13 , 26 , 39 , 52 , 65 , 78

6 times 13

7 will be more than this

Least number is 6 for 13

78! can be divided by 91⁶