Grade what is the units digit of the simplest form of ? Answer: Solution

GRADE 8

1. 
   What is the units digit of the simplest form of ?
   

Answer: 1
Solution:

ends with 1

2. 
   What is the highest power of 200 in 90!?
   

Answer: 10
Solution:
200 = 2³ x 5²
The highest power of prime p^a in n! is where p is prime
Highest power of 2³ is = 26
Highest power of 5² is = 10
Highest power of 200 in 90! is 10

3. 
   What is the two rightmost digits of 8⁵⁸¹?
   

Answer: 08
Solution:
2³⁽⁵⁸¹⁾ = (2¹⁰)¹⁷⁴ x 2³  (24)¹⁷⁴ x 2³  76 x 8  08

4. 
   How many numbers from 1 to 1000 are relatively prime to either 6 or 35 (not both)?
   

Answer: 790
Solution:
Relatively prime to 6 ; 1000 (1-1/2)(1-1/3)=333
Relatively prime to 35; 1000 (1-1/5) (1-1/7) = 685
Relatively prime to both 6 and 35; 1000 (1-1/2)(1-1/3) (1-1/5)(1-1/7) =228
Relatively prime to either 6 or 35 = 333 + 685 - 228 = 790

5. 
   There are m ordered pairs (a, b) that satisfy the equation ab=728. If a and b are integers, what is m?
   

Answer: 32
Solution: 728 = 2³ x 7 x 13
There are 4 x 2 x 2 = 16 factors
There are 16 pairs of positive ordered pairs whose product is 728.
The negative pairs will also give a positive product, thus there are 16 x 2 =32 pairs.

6. 
   How many numbers that divides 81,000 are perfect square?