Question

every visitor who visits a travel fair receives a free travel bag. the travel company puts a pen in every 2nd bag it packs. it puts a keychain in every 6th bag and a discount voucher in every 8th bag. how many of the first 80 travel bags will contain all the 3 items (a pen, a keychain and a discount voucher)?

Explanation:

Step1: Find LCM of 2, 6, 8

We need to find the least common multiple (LCM) of 2, 6, and 8. Prime factorize each number:

• \(2 = 2\)
• \(6 = 2\times3\)
• \(8 = 2^3\)

LCM is the product of the highest powers of all primes: \(2^3\times3 = 24\). So every 24th bag will have all three items.

Step2: Calculate number of such bags in 80

We divide 80 by 24 and take the quotient (since we want the number of complete cycles of 24).
\(80\div24 = 3\) with a remainder of 8.

Answer:

3