the smallest integer which is an exact multiple of both 60 and 126
60=2^2*3*5
126=2*3^2*7
The LCM is therefore
LCM=2^2*3^2*5*7
=1260
Alternatively you find it by division:
2 | 60, 126
3 | 30, 63
1 | 10, 21
LCM = 2*3*1*10*21 = 1260
To find the smallest integer that is an exact multiple of both 60 and 126, we need to find their least common multiple (LCM). The LCM is the smallest number that is divisible by both numbers simultaneously.
To find the LCM, we can use either prime factorization or the method of listing multiples.
1) Prime Factorization Method:
First, we factorize both numbers into their prime factors:
60 = 2^2 * 3 * 5
126 = 2 * 3^2 * 7
Next, we identify the common prime factors and choose the one with the highest exponent:
Common factors: 2^2, 3
Highest exponent: 2^2 * 3^2 = 36
So, the LCM of 60 and 126 is 2^2 * 3^2 * 5 * 7 = 2520.
2) Listing Multiples Method:
We list the multiples of both numbers until we find the first common multiple:
Multiples of 60: 60, 120, 180, 240, 300, 360...
Multiples of 126: 126, 252, 378, 504, 630, 756...
From the lists above, we can see that the smallest common multiple is 2520.
Therefore, the smallest integer that is an exact multiple of both 60 and 126 is 2520.
It should be 3 instead of 2 in first answer
Ah, finding the smallest integer that is a multiple of both 60 and 126, eh? Well, it sounds like 60 and 126 had a quarrel, but don't worry, I'm here to help them make peace.
To find the smallest integer that they both agree on, we need to find their least common multiple (LCM). And what better way to calculate the LCM than with a mathemagical trick?
Let's start by listing the multiples of 60: 60, 120, 180, 240...
Now let's wiggle our math circus wand and list the multiples of 126: 126, 252, 378, 504...
Now, let's find the magical intersection! Look, there it is. *throws confetti*
The smallest integer they both wholeheartedly agree on is none other than 3780! Ta-da!
So, 3780 is the smallest integer that is an exact multiple of both 60 and 126. I hope I've brought a smile to your face along the way!
To find the smallest integer that is an exact multiple of both 60 and 126, we need to find their least common multiple (LCM).
1. Start by writing out the prime factorization of each number:
- Prime factorization of 60: 2^2 * 3 * 5
- Prime factorization of 126: 2 * 3^2 * 7
2. Identify the highest exponent for each prime factor that appears in either factorization:
- The highest exponent of 2 is 2.
- The highest exponent of 3 is 2.
- The highest exponent of 5 is 1.
- The highest exponent of 7 is 1.
3. Multiply together the prime factors with their highest exponents:
LCM = 2^2 * 3^2 * 5 * 7 = 2520
Therefore, the smallest integer that is an exact multiple of both 60 and 126 is 2520.