Question

The number of zeroes in 140! ........

Please notice it's 140 factorial

Answer 1

Answer : The number of trailing 0s in 140! is 34

Detail Explanation

You get zeros at the end of an answer whenever you multiply by a number with a factor of ten, or whenever you multiply by numbers which give you a 2 and a 5 that work together as a pair...

Step 1: How many factors of 10?

A) There are 14 numbers which have at least one factor of 10. (10, 20, 30, . . . 140)

B) There is one number which has ANOTHER factor of 10. (100)

Therefore, there are 15 factors of "10" to be found. However, WE ARE GOING TO THROW THESE 16 FACTORS AWAY! (Why? Because they're all going to be counted again in Steps 2 and 3...)


Step 2: How many factors of 2?

A) There are 70 numbers that have at least one factor of 2. (2, 4, 6, 8, . . . 136, 138, 140)

B) There are 35 numbers that have at least a SECOND factor of 2. (4, 8, 12, 16, . . . 136, 140)

C) There are 17 numbers that have at least a THIRD factor of 2. (8, 16, 24, 32, . . . 136)

D) There are 8 numbers that have at least a FOURTH factor of 2. (16, 32, 48, . . . 128)

E) There are 4 numbers that have at least a FIFTH factor of 2. (32, 64, 96, 128)

F) There are 2 numbers that have at least a SIXTH factor of 2. (64, 128)

G) There is 1 number that have a SEVENTH factor of 2. (128)


This gives us a total of 137 factors of two in 150! However, these will only give us a zero in the answer if they get paired with a "5." So -- let's search for fives...


Step 3: How many factors of 5?

A) There are 28 numbers that have at least one factor of 5. (5, 10, 15, 20, . . . 140)

B) There are 5 numbers that have at least TWO factors of 5. (25, 50, 75, 100, 125)

C) There is 1 number that has THREE factors of 5. (125)


This gives us 34 factors of 5. Paired with 34 of the 146 factors of 2, that will give us 34 zeros at the end of the standard form of 150! (The other twos are nice and all, but won't give us any more zeros. At least, not at the END...)

Notice -- all those numbers which were multiples of 10 were in fact counted in Steps 2 and 3. That's why we could talk about them in Step 1, but we didn't need to COUNT them at that point...