Question

Two numbers, both greater than 23, have HCF 23 and LCM 644. The sum of the numbers is:

Answer 1

Answer: 253

Step-by-step explanation:

First, we have two numbers A and B.

We know that all numbers can be written as a product of prime numbers.

And the HCF of A and B is 23

Then we can write:

A = 23*a

B = 23*b

(remember that 23 is a prime number)

Where a and b are larger than 1.

Now we know that the least common multiple between them is 644.

Notice that the primes that conform a can not be the same as the ones for b, because in that case, the HCF would not be 23.

Now to find a and b we can compute:

644/23 = 28.

28 = a*b

Now let's write 28 as a product of prime numbers:

a*b = 28 = 4*7 = 2*2*7

Then we can define:

a = 2*2

b = 7.

Our two numbers are:

A = 23*2*2 = 92

B = 23*7 = 161

A + B = 92 + 161 = 253

Answer 2

The sum of the two numbers would be:

253

Assume the two numbers be $x$ and $y$

Given that,

HCF of $x$ and $y$ = 23

so,

The numbers could be written as:

23$x$ and 23$y$

We know that,

LCM of 23$x$ and 23$y$ = 644

We also know that,

LCM × HCF = The product of two numbers

∵ 644 × 23 = 23$x$ × 23$y$

So, by solving them we get;

14812/529 = $x$ × $y$

∵ $x$ × $y$ = 28

If we prime factorize 28, we get

28 = 2 × 2 × 7

This gives us,

$x =$23 × 2 × 2 = 92

$y =$ 23 × 7 = 161

Thus, the sum of the two numbers = 92 + 161 = 253

Learn more about "HCF" here:

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