Question

Find the hcf and lcm 25152 of and 12156 by using the fundamental theorem of arithmetic

Answer 1

Approach 1. Integer numbers prime factorization:

25,152 = 26 × 3 × 131;

12,156 = 22 × 3 × 1,013;

Take all the prime factors, by the largest exponents.

Least common multiple

lcm (25,152; 12,156) = 26 × 3 × 131 × 1,013 = 25,478,976;

Least common multiple, lcm (48,624; 12,156) = ?

Approach 2. Euclid's algorithm:

Calculate the greatest (highest) common factor (divisor), gcf (gcd), gcf, gcd:

Step 1. Divide the larger number by the smaller one:

25,152 ÷ 12,156 = 2 + 840;

Step 2. Divide the smaller number by the above operation's remainder:

12,156 ÷ 840 = 14 + 396;

Step 3. Divide the remainder from the step 1 by the remainder from the step 2:

840 ÷ 396 = 2 + 48;

Step 4. Divide the remainder from the step 2 by the remainder from the step 3:

396 ÷ 48 = 8 + 12;

Step 5. Divide the remainder from the step 3 by the remainder from the step 4:

48 ÷ 12 = 4 + 0;

At this step, the remainder is zero, so we stop:

12 is the number we were looking for, the last remainder that is not zero.

This is the greatest common factor (divisor).

Least common multiple:

lcm (a; b) = (a × b) / gcf, gcd (a; b);

lcm (25,152; 12,156) = (25,152 × 12,156) / gcf, gcd (25,152; 12,156) = 305,747,712 / 12 = 25,478,976;

Least common multiple

lcm (25,152; 12,156) = 25,478,976 = 26 × 3 × 131 × 1,013;

Final answer:

Least common multiple

lcm (25,152; 12,156) = 25,478,976 = 26 × 3 × 131 × 1,013; .

Answer 2

21252 or 8232 hcf

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