What is the remainder when 51 × 27 × 35 × 62 × 75 is divided by 100?

The correct option is 50.

To find the remainder when the product

$P=51\times 27\times 35\times 62\times 75$

is divided by 100, we can simplify the expression by finding common factors with the divisor 100.

Step 1: Simplify the product and divisor
Let us write the expression for the remainder as:

$R=(51\times 27\times 35\times 62\times 75)\mathrm{mod}100$

Notice that numbers in the product share common factors with 100.
For example, 75 and 100 are both divisible by 25:

$\frac{75}{25}=3$

$\frac{100}{25}=4$

So, we cancel out a factor of 25 from the numerator (product) and the denominator (divisor 100).
After dividing both by 25, the problem reduces to finding the remainder when the remaining product is divided by 4.

Step 2: Calculate the reduced product modulo 4
The reduced product is:

$P^{\prime }=51\times 27\times 35\times 62\times 3$

Now, we find the remainder of each term when divided by 4:

$51\mathrm{mod}4=3\equiv -1\left(\mathrm{mod}4\right)$

$27\mathrm{mod}4=3\equiv -1\left(\mathrm{mod}4\right)$

$35\mathrm{mod}4=3\equiv -1\left(\mathrm{mod}4\right)$

$62\mathrm{mod}4=2$

$3\mathrm{mod}4=3\equiv -1(\mathrm{mod<mspace\; width="mediummathspace"></mspace><mn>4</mn><mo>)</mo>}$

Substituting these remainders into the reduced product:

$P^{\prime }\mathrm{mod}4\equiv (-1)\times (-1)\times (-1)\times 2\times (-1)(\mathrm{mod<mspace\; width="mediummathspace"></mspace><mn>4</mn><mo>)</mo>}$

$P^{\prime }\mathrm{mod}4\equiv 1\times 2=2$

So, the remainder when the reduced product is divided by 4 is 2.

Step 3: Restore the original remainder
Since we initially divided the whole expression (both the product and the divisor 100) by a factor of 25, we must multiply the reduced remainder by 25 to get the original remainder:

$\text{Original Remainder}=2\times 25=50$

Thus, the remainder when 51 × 27 × 35 × 62 × 75 is divided by 100 is 50.