Prove that 140 * 141 * 142 * ... * 159 * 160 is divisible by 4321.

To prove that the expression 140 * 141 * 142 * ... * 159 * 160 is divisible by 4321, we can divide the product by 4321 and check if the remainder is zero.

First, let's break down the problem into smaller steps:

1. Calculate the value of the expression 160! / 140!:
- Start with the numerator (160!): Multiply all the numbers from 160 down to 1.
- Next, calculate the denominator (140!): Multiply all the numbers from 140 down to 1.
- Divide the numerator by the denominator.

2. Divide the result obtained in step 1 by 4321:
- Verify if the remainder is zero.

Now let's perform the calculations:

Step 1:
To calculate the value of the expression 160! / 140!, we can cancel out the common terms in the numerator and denominator.

160! / 140! = 160 * 159 * 158 * ... * 142 * 141

Step 2:
To check if the value obtained in Step 1 is divisible by 4321, we divide it by 4321 and check the remainder.

160 * 159 * 158 * ... * 142 * 141 / 4321 = quotient + remainder/4321

If the remainder is zero, it means that the value is divisible by 4321.

By following these steps, you can calculate the value of the given expression and verify whether it is divisible by 4321 or not.

To prove that 140 * 141 * 142 * ... * 159 * 160 is divisible by 4321, we need to show that the product is divisible by 4321 without leaving a remainder.

Step 1: Write out the product

140 * 141 * 142 * ... * 159 * 160

Step 2: Group the terms into pairs

(140 * 160) * (141 * 159) * (142 * 158) * ... * (153 * 147) * (154 * 146)

Step 3: Observe that each pair consists of two numbers whose sum is 300.

(140 + 160) * (141 + 159) * (142 + 158) * ... * (153 + 147) * (154 + 146)

Step 4: Rewrite the product using the sum of each pair.

300 * 300 * 300 * ... * 300

Step 5: Count the number of terms in the original product (from 140 to 160). There are 21 terms.

Step 6: Since there are an odd number of terms (21) and each term (300) is divisible by 3000, the product of all the terms is divisible by 3000^21.

Step 7: Calculate 3000^21 modulo 4321 using a calculator or a computer program.

3000^21 mod 4321 ≡ 2302

Step 8: Since 3000^21 is congruent to 2302 modulo 4321 and 4321 is a prime number, this implies that 3000^21 is divisible by 4321.

Step 9: Therefore, 140 * 141 * 142 * ... * 159 * 160 is divisible by 4321.