Large Numbers Around Us - Patterns in Products

Find the product of $11111 \times 11111$ using patterns.

$11111 \times 11111 = 123454321$

Since there are $5$ ones in the number, the pattern starts from $1$, goes up to $5$, and then decreases back to $1$.

Calculate $999 \times 456$ using the distributive property pattern.

$999 \times 456 = (1000 - 1) \times 456$ $\text{= } (1000 \times 456) - (1 \times 456)$ $\text{= } 456000 - 456$ $\text{= } 455544$

By expressing $999$ as $(1000 - 1)$, we convert a complex multiplication into a simple subtraction of a large number.

Solve the following vertical multiplication: $4000 \times 250$.

$\begin{array}{r} 4000 \\ \times 250 \\ \hline 1000000 \end{array}$

First, multiply the non-zero digits: $4 \times 25 = 100$. Then, count the total number of trailing zeros in both factors ($3$ zeros from $4000$ and $1$ zero from $250$, total $4$). Append $4$ zeros to $100$ to get $1,000,000$.

Find the square of $65$ using the pattern for numbers ending in $5$.

$65 \times 65 = (6 \times 7) \text{ concatenated with } 25$ $\text{= } 4225$

For any number $n5$, the product is found by multiplying the digit $n$ by its successor $(n+1)$ and placing $25$ at the end. Here $n=6$, so $6 \times 7 = 42$.