Find the greatest number which when divided into 179 and 234 will leave a remainder of 3 in each case
179 = 176+3
234 = 231+3
176 = 16*11
231 = 21*11
So 11 is the greatest common factor.
Well, finding the greatest number that satisfies this condition requires a bit of cleverness. Let's take a look at the problem.
If a number divides into 179 and leaves a remainder of 3, it means that the number can be written as 179k + 3. Similarly, if the same number divides into 234 and leaves a remainder of 3, it can be written as 234n + 3.
So, we need to find a number that satisfies both equations: 179k + 3 = 234n + 3.
Subtracting 3 from both sides, we have: 179k = 234n.
Now, let's simplify this equation a bit. We can rewrite 179 as (2*89), and 234 as (2*3*3*13). So the equation becomes:
2 * 89 * k = 2 * 3 * 3 * 13 * n.
Considering that 2 is a common factor on both sides, we can cancel it out:
89 * k = 3 * 3 * 13 * n.
Now, we see that 89 is a prime number, and 3, 13 are also primes. So, in order for this equation to be satisfied, k must contain 3, 3, and 13 as factors.
Therefore, the greatest number that satisfies this condition is the product of 3, 3, and 13, which is 3 * 3 * 13 = 117.
So, the answer is 117. And just for fun, I'll throw in a clown joke: Why don't clowns ever ride bicycles? Because they always joke around!
To find the greatest number that leaves a remainder of 3 when divided into 179 and 234, we need to find the greatest common divisor (GCD) of these two numbers minus 3.
Step 1: Subtract 3 from both numbers.
179 - 3 = 176
234 - 3 = 231
Now we need to find the GCD of 176 and 231.
Step 2: Divide the larger number by the smaller number.
231 ÷ 176 = 1 with a remainder of 55
Step 3: If there is a remainder, divide the smaller number by the remainder.
176 ÷ 55 = 3 with a remainder of 11
Step 4: Repeat steps 2 and 3 until there is no remainder.
55 ÷ 11 = 5 with no remainder
Step 5: The last divisor (11) is the GCD.
Therefore, the greatest number that leaves a remainder of 3 when divided into 179 and 234 is 11.
To find the greatest number that leaves a remainder of 3 when divided into 179 and 234, we can use the concept of the greatest common divisor (GCD) or highest common factor (HCF).
The remainder when a number is divided by another number can also be expressed as the difference between the two numbers. In this case, the remainder of 3 when divided into 179 can be written as 179 - 3 = 176, and the remainder of 3 when divided into 234 can be written as 234 - 3 = 231.
So, we are looking for the greatest number that divides both 176 and 231. To find this number, we need to calculate the GCD/HCF of 176 and 231.
One way to do this is using the Euclidean algorithm. Here's how it works:
1. Divide the larger number by the smaller one: 231 ÷ 176 = 1 remainder 55.
2. Now, divide the smaller number (176) by the remainder (55): 176 ÷ 55 = 3 remainder 11.
3. Repeat step 2 with the new remainder: 55 ÷ 11 = 5 remainder 0.
Since the remainder is now 0, the last non-zero remainder obtained (11) is the GCD of 176 and 231.
Therefore, the greatest number that leaves a remainder of 3 when divided into 179 and 234 is 11.