Smallest number divisible by 35, 32, 40, and 80

To find the smallest number that is divisible by 35, 32, 40, and 80, we need to find their least common multiple (LCM). The LCM is the smallest positive integer that is divisible by all the given numbers.

Step 1: Prime factorization
- Prime factorize each number individually to find their prime factors.

35: 5 × 7
32: 2^5
40: 2^3 × 5
80: 2^4 × 5

Step 2: Identify the common and uncommon prime factors
- Look for the prime factors that are common to all the numbers.

35: 5 × 7
32: 2^5
40: 2^3 × 5
80: 2^4 × 5

The common prime factors are 2, 5, and none for 7.

Step 3: Determine the highest exponent
- Find the highest exponent for each common prime factor.

2: highest exponent is 5
5: highest exponent is 1

Step 4: Calculate the LCM
- Multiply the common prime factors with their highest exponents to calculate the LCM.

LCM = 2^5 × 5^1 = 32 × 5 = 160

Therefore, the smallest number that is exactly divisible by 35, 32, 40, and 80 is 160.

Summary:
- The prime factorization of each number is:
- 35: 5 × 7
- 32: 2^5
- 40: 2^3 × 5
- 80: 2^4 × 5
- The common prime factors are 2 and 5.
- The highest exponent for 2 is 5 and for 5 is 1.
- The LCM is calculated by multiplying the common prime factors with their highest exponents: LCM = 2^5 × 5^1 = 160.
- Therefore, the smallest number that is divisible by 35, 32, 40, and 80 is 160.

5 is the right answer