Ex 9.1, 15

Transcript

Ex 9.1, 15 Consider the statement: ‘If a number is divisible by 8, then it is divisible by both 2 and 4’. Justify the statement. Recall the divisibility shortcuts that we have studied for different numbers. To check whether a given number is divisible by 8, is it enough to check whether it is divisible by 2 and 4? Why or why not? Let’s do this one by one Statement: If a number is divisible by 8, then it is divisible by both 2 and 4. (i) Justify the statement. Let number be n Since number is divisible by 8, we can write 𝒏=𝟖𝒌 for an integer 𝑘 Now, 𝒏=8𝑘=𝟐 × (𝟒𝒌) Thus, n is divisible by 2 Also: 𝒏=8𝑘=𝟒 × (𝟐𝒌) Thus, n is divisible by 4 Therefore every number divisible by 8 is divisible by both 2 and 4 Hence justified (ii) Is checking divisibility by 2 and 4 enough to check divisibility by 8? This is not true Example, take number as 12 Now, 12/2=6 12/4=3 But 𝟏𝟐/𝟖=𝟏.𝟓 Thus, 12 is not divisible by 8 . So, we can say divisibility by both 2 and 4 does not guarantee divisibility by 8 . The correct shortcut is to check whether the number formed by the last three digits is divisible by 8. For a number with fewer than three digits, check the number itself. Since number is divisible by 8, we can write 𝒏=𝟖𝒌 for an integer 𝑘 Now, 𝒏=8𝑘=𝟐 × (𝟒𝒌) Thus, n is divisible by 2 Also: 𝒏=8𝑘=𝟒 × (𝟐𝒌) Thus, n is divisible by 4 Therefore every number divisible by 8 is divisible by both 2 and 4 Hence justified (ii) Is checking divisibility by 2 and 4 enough to check divisibility by 8? This is not true Example, take number as 12 Now, 12/2=6 12/4=3 But 𝟏𝟐/𝟖=𝟏.𝟓 Thus, 12 is not divisible by 8 . So, we can say divisibility by both 2 and 4 does not guarantee divisibility by 8 . The correct shortcut is to check whether the number formed by the last three digits is divisible by 8. For a number with fewer than three digits, check the number itself. (ii) Is checking divisibility by 2 and 4 enough to check divisibility by 8? This is not true Example, take number as 12 Now, 12/2=6 12/4=3 But 𝟏𝟐/𝟖=𝟏.𝟓 Thus, 12 is not divisible by 8 . So, we can say divisibility by both 2 and 4 does not guarantee divisibility by 8 . The correct shortcut is to check whether the number formed by the last three digits is divisible by 8. For a number with fewer than three digits, check the number itself.