*Index of Blogs*

The following puzzles are based on some interesting situations that use

*least common multiple*(LCM)

*of a set of numbers.*

LCM
of a set of numbers is the smallest number that is completely divisible by each
of them without a remainder.

(you can use a calculator if you wish; Answers at the end)

**Puzzle 1:**In a party, the host wants to arrange exactly the same number of guests at each table. He tries to seat 4, 5 and 6 guests per table but finds that, in each case, the last table has only 3 guests. However, if he tries 7 guests per table then they fit exactly. How many guests did the host invite?

Some variations of the above puzzle, slightly more difficult, are:

**Puzzle 2:**In a party, the host wants to arrange exactly the same number of guests per table. He tries to seat 4, 5 and 6 guests on each table but finds that, in each case, the last table has only 2 guests. However, if he tries 7 guests per table then they fit exactly. How many guests did the host invite?

**Puzzle 3:**In a party, the host wants to arrange exactly the same number of guests per table. He tries to seat 4, 5 and 6 guests on each table but finds that, in each case, the last table has only 1 guest. However, if he tries 7 guests per table then they fit exactly. How many guests did the host invite?

Moving on to a different situation, consider the following puzzle

**Puzzle 4:**Jack and Debbie want to combine their substantial stamp collections and prepare a new folder. Sticking only odd number of stamps in a row, they find that when they use

5
stamps in a row then 4 stamps are left in the last row

7
stamps in a row then 6 stamps are left in the last row

9
stamps in a row then 8 stamps are left in the last row

11
stamps complete all the rows perfectly with none left over.

How
many stamps in total do they have?

**Puzzle 5:**The priest wishes to arrange his congregation is rows such that all rows have exactly the same number of people. He does not like 13 people, not an auspicious number, in a row.

He
finds that if he sits them in rows of 7, 8, 9, 10, 12, 14, 15 or 16 then he is
left with 6, 7, 8, 9, 11, 13, 14 and 15 in the last row. Not what he wants, but rows of 11 fill
properly.

How
many people are there in the congregation?

**Answers:**

1. 63; 2. 182; 3. 301; 4 and 5. 2519