11. The height of a slippery pole is 10 m and an insect is trying to climb the pole. The insect climbs 5 m in one minute and then slips down by 4 m. In how much time will the insect reach the top?
12. Which is greater, the sum of the first twenty whole numbers or the product of the first twenty whole numbers?
13. If a whole number is divisible by 2 and 4, is it divisible by 8 also?
Answer (2)
The insect takes a total of 6 minutes to reach the top of the 10 m pole. It climbs 5 m in the first minute, then slips down 4 m, and continues this cycle until it reaches the top. In the 6th minute, it climbs the remaining 5 m without slipping back.
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Let's address each part of the student's question step-by-step.
Insect Climbing the Pole
The insect climbs 5 meters in one minute and slips down 4 meters, resulting in a net climb of 1 meter per minute. The pole is 10 meters tall, so:
Net Climb per Minute : 5 − 4 = 1 meter
The insect climbs 9 meters in the first 9 minutes (9 minutes x 1 meter per minute).
On the 10th minute, it climbs the last 5 meters to reach the top, without slipping back.
Therefore, the insect reaches the top in 10 minutes.
Sum vs. Product of the First Twenty Whole Numbers
Sum of the First 20 Whole Numbers :
The formula for the sum of the first n whole numbers is 2 n ( n + 1 ) .
For the first 20 whole numbers: 2 20 × 21 = 210
Product of the First 20 Whole Numbers :
The product is 1 × 2 × 3 × … × 20 , which is known as 20! (factorial).
Even without calculating the exact value, since 20! involves multiplying numbers, it is significantly larger than 210.
Therefore, the product of the first 20 whole numbers is greater than their sum.
Divisibility Question
If a number is divisible by both 2 and 4, it must be an even number and a multiple of 4.
For a number to be divisible by 8, it should be a multiple of 8.
A number divisible by both 2 and 4 is sometimes divisible by 8, but not always. An example is 12, which is divisible by 2 and 4, but not 8.
Therefore, a whole number divisible by 2 and 4 is not necessarily divisible by 8.
