Find the greatest number that can be exactly divisible by 147, 161 ,210
The Correct Answer and Explanation is:
To find the greatest number that can be exactly divisible by 147, 161, and 210, we need to find the Least Common Multiple (LCM) of these three numbers. The LCM of multiple numbers is the smallest number that can be evenly divided by each of them.
Step 1: Prime Factorization
Let’s begin by finding the prime factorizations of each number:
- 147:
147 is divisible by 3 (since the sum of digits, 1 + 4 + 7 = 12, is divisible by 3).
147÷3=49147 ÷ 3 = 49147÷3=49
49 is divisible by 7.
49÷7=749 ÷ 7 = 749÷7=7
So, 147=3×72147 = 3 \times 7^2147=3×72. - 161:
161 is divisible by 7.
161÷7=23161 ÷ 7 = 23161÷7=23
23 is prime.
So, 161=7×23161 = 7 \times 23161=7×23. - 210:
210 is divisible by 2 (since it is an even number).
210÷2=105210 ÷ 2 = 105210÷2=105
105 is divisible by 3 (sum of digits, 1 + 0 + 5 = 6, divisible by 3).
105÷3=35105 ÷ 3 = 35105÷3=35
35 is divisible by 5.
35÷5=735 ÷ 5 = 735÷5=7
So, 210=2×3×5×7210 = 2 \times 3 \times 5 \times 7210=2×3×5×7.
Step 2: Find the LCM
To find the LCM, we take the highest powers of all primes that appear in the factorizations:
- The prime factors involved are 2, 3, 5, 7, and 23.
- The highest powers of each are:
- 212^121 (from 210),
- 313^131 (from 147 and 210),
- 515^151 (from 210),
- 727^272 (from 147),
- 23123^1231 (from 161).
Thus, the LCM is:LCM=21×31×51×72×231LCM = 2^1 \times 3^1 \times 5^1 \times 7^2 \times 23^1LCM=21×31×51×72×231
Step 3: Calculate the LCM
Now, calculate the value:LCM=2×3×5×49×23LCM = 2 \times 3 \times 5 \times 49 \times 23LCM=2×3×5×49×23=2×3=6= 2 \times 3 = 6=2×3=66×5=306 \times 5 = 306×5=3030×49=147030 \times 49 = 147030×49=14701470×23=33,8101470 \times 23 = 33,8101470×23=33,810
Final Answer:
The greatest number that can be exactly divisible by 147, 161, and 210 is 33,810.
Explanation:
The process involves finding the prime factors of each number, then selecting the highest powers of all the primes that appear in any of the factorizations. This gives the LCM, which is the smallest number divisible by each of the given numbers. The LCM ensures that all the numbers divide evenly into it, providing the solution to the problem.
