JAMIA 2023 — Mathematics PYQ

Step 1: Observe the pattern of factorials

A factorial $n!$ ends with two or more zeros if it is divisible by $100$.

Since $10!=10\times 9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1$, it contains the factors $10$, $5$, and $2$ ($10\times 5\times 2=100$).

Therefore:

This means we only need to calculate the sum of factorials from $1!$ to $9!$ modulo $100$.

Step 2: Calculate factorials modulo 100

• $1!=1$

• $2!=2$

• $3!=6$

• $4!=24$

• $5!=120\equiv 20(\mathrm{mod}100)$

• $6!=720\equiv 20(\mathrm{mod}100)$

• $7!=5040\equiv 40(\mathrm{mod}100)$

• $8!=40320\equiv 20(\mathrm{mod}100)$

• $9!=362880\equiv 80(\mathrm{mod}100)$

Step 3: Find the sum

Summing these values:

Final Conclusion:

The last two digits of the sum are $13$.

• The units digit is $3$.

• The tens digit is $1$.

Final Answer:

The tens digit is $1$.