Our class planned a holiday party for disadvantaged kids.Some of us baked cookies for the party.On the day of the party,we found we could divide the cookies into packets of two, three, four, five, or six and have just one cookie left over in each case.If we divided them into packets of seven, there would be no cookies left over. What is the least number of cookies the class could have baked?__________
Start with the multiples of 7 and then figure which is just one more than the other multiples.
301
To find the least number of cookies the class could have baked, we need to look for the smallest number that satisfies the given conditions.
We are told that when dividing the cookies into packets of two, three, four, five, or six, there is always one cookie left over. This means that the number of cookies must have a remainder of 1 when divided by 2, 3, 4, 5, and 6.
To find the least number that satisfies these conditions, we can start by looking for the least common multiple (LCM) of 2, 3, 4, 5, and 6.
The LCM of 2, 3, 4, 5, and 6 is 60.
However, we also know that when dividing the cookies into packets of seven, there would be no cookies left over. This means that the number of cookies must be a multiple of 7.
To find the least common multiple of 60 and 7, we can multiply these numbers together:
LCM (60, 7) = 420.
Therefore, the least number of cookies the class could have baked is 420.
To find the least number of cookies the class could have baked, we need to find the smallest number that satisfies all the given conditions.
We are told that the class can divide the cookies into packets of 2, 3, 4, 5, or 6 with one cookie left over in each case. This means that the number of cookies must be one more than a multiple of 2, 3, 4, 5, and 6. We can write this as the following equations:
Number of cookies ≡ 1 (mod 2)
Number of cookies ≡ 1 (mod 3)
Number of cookies ≡ 1 (mod 4)
Number of cookies ≡ 1 (mod 5)
Number of cookies ≡ 1 (mod 6)
To find the least number that satisfies all these equations, we can find the least common multiple (LCM) of 2, 3, 4, 5, and 6, and then add 1 to it.
The LCM of 2, 3, 4, 5, and 6 is 60. Adding 1 to it, we get 61.
Therefore, the least number of cookies the class could have baked is 61.