6AM Quiz (2011, 16 July): Huge numbers are no big deal!

Which number is larger:
(987 654 324)^2 
or
(987 654 320) x (987 654 328) ?
Justify your answer without the use of calculator.

27 comments:

  1. As the 8 digits of all 3 numbers are the same, we only see the last digit which is different. So, the 'blue' number is 4*4=16 while the pink is 0*8=0.
    And thus, the 'blue' number is larger than the 'pink' number.

    Owen here ._.

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  2. @Owen:
    Interesting way of explaining the solution.
    You are using one example to show that your case is true.
    Is what you explain true for all cases? or what you explain is an 'exceptional' case?

    Hint: Would algebra helps to explain? :P

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  3. The pink number can also be written as (987 654 324) x (987 654 324) . So now , the only difference between the two pairs of numbers is their last digit .
    So , we will take 4*4 and 0*8 to see which pair of numbers is larger .
    4*4 = 16 <- Larger .
    8*0 = 0
    so the answer is the pink number .

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  4. @QianHui:
    Yes, there's obvious pattern as you observed :)
    On the other hand, do you think this works for other sets of numbers, too?

    Hint: Would algebra helps to explain? :P

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  5. The only difference between the 2 numbers is the last digit.
    (987 654 324)^2 -> last digit is 4
    (987 654 320) x (987 654 328) -> last digit is 8 and 0
    4^2 = 16
    8*0 = 0
    16>0
    Therefore, 987 654 324)^2 is the greater number.

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  6. This comment has been removed by the author.

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  7. @Owen @QianHui @Sandy :

    Does what you say still stand if I change the pair of numbers to
    (987 654 320)^2
    and
    (987 654 316) x (987 654 324) ?

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  8. In both of the 2 numbers, all the numbers except the last digit is the same.
    So, (987 654 324)^2=(987 654 324)(987 654 324)
    (987 654 320) x (987 654 328)
    so if you take out all the same digits (all except the last digit).
    You will have 4x4=16 and 8x0=0
    and 16 is bigger than 0
    so (987 654 324)^2 is bigger than (987 654 320) x (987 654 328)

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  9. @ Ms Loh: and yes i will stand still

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  10. @Ms Loh:

    Now its the last two digit. 24 x 16 = 384 which 20 x 20 = 400 So the 1st one is still the larger one!

    Owen here.

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  11. The digits of all the numbers are the same except for the last digit which is 4 for the squared number, and 0 and 8 for the other number.

    Thus, 4^2 is 4 x 4 = 16

    While 0 x 8 = 0

    So the difference is 16, so the value of (987 654 324)^2 is bigger than (987 654 320) x (987 654 328)

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  12. Take 987 654 320 as a.
    Then 987 654 324=a+4
    and 987 654 328=a+8.

    987 654 324^2=(a+4)^2
    =(a+4)(a+4)
    =a^2+8a+16

    987 654 320*987 654 328=a(a+8)
    =a^2+8a

    Thus the blue one is bigger than the pink one by 16.

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  13. The only difference between the two numbers is the unit digit.
    To find out the bigger number, we multiply the unit digits :

    'Blue' number : 4 ^ 2 = 16 ( Bigger number)
    'Pink' number : 8 * 0 = 0

    Thus, the 'Blue' number is bigger.

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  14. The digits are all the same, except the last digit, so we take out their last digits.

    4x4=16 8x0=0

    16-0=16

    The blue one is bigger than the pink one by 16.

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  15. The 1st number in blue can also be written as 987 654 324*987 654 324, so the only difference between the blue numbers and the pink number are their last digit.

    Blue numbers: 4*4=16
    Pink numbers: 8*0=8

    Thus the blue one is larger than the pink one.

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  16. I think that (987 654 324)^2 is larger than (987 654 320) x (987 654 328).
    Without using a calculator
    24^2=24*24
    =576
    and
    20*28=560
    Because 24+24=48 and 20+28=48
    If the last two digits of the blue number are smaller(example: 23^2), then the pink number would be larger.
    23*23= 529(not using calculator)
    If the last two digits of the blue number are bigger(example:26*26), the blue number would still be larger.
    26*26=676(not using calculator)

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  17. (987 654 324)^2 is 987 654 324 * 987 654 324

    The difference between 987 654 324 * 987 654 324 and 987 654 320 x 987 654 328 is the last digit.
    4*4 = 16
    8*0 = 0
    So (987 654 324)^2 , the blue number is larger.
    Gladys.

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  18. We can take the smallest value among the two equation which is 987 654 320 to be x.

    Let 987654320 be x.
    987 654 324 = (x+4)
    987 654 328 = (x+8)

    (987 654 324)^2 can be written as (x+4)^2 in the algebraic method.

    (x+4)^2= (x+4)(x+4)
    = x^2+4x+4x+16
    =x^2+8x+16

    (987 654 320) x (987 654 328) can be written as (x)(x+8)

    (x)(x+8)= x^2+8x

    (x^2+8x+16) is larger than x^2+8x so (987 654 324)^2 is larger than (987 654 320) x (987 654 328).

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  19. let 98754320 be X

    (987 654 324)^2
    = (X+4)^2
    = (X)^2 + 2(X)(4) + (4)^2
    = X^2 + 8X +16

    (987 654 320) x (987 654 328)
    = (X)(X+8)
    =X^2 +8X

    (987 654 324)^2 > (987 654 320) x (987 654 328)

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  20. @Nina @Mavis @Yi Chieh

    Yes, I was waiting for someone tonexplainnthe 'phenomenon' using algebra so that the way to explain similar kind of problem can be genralised :)

    BTW, it's seems that all 3 of you choose to let the variable be 987 654 320.
    Can we choose to substitute it with 987 654 324?

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  21. The only difference between the numbers are the last digits. So we take the last digits and form:
    4*4= 16
    0*8= 0
    (987 654 324)^2 > [(987 654 320) x (987 654 328)]

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  22. @ MS Loh

    Yes, substituting 987 654 324 with a is also possible and we will get the same answer. Actually, that was my initial way of answering the question. :)

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  23. (987 654 324)^2
    or
    (987 654 320) x (987 654 328)

    (987 654 324)^2
    =(987 654 324)(987 654 324)
    =987 654 320^2+4*4

    (987 654 320)(987 654 328)
    =987 654 320^2+0*8

    So basically you're comparing 4*4 to 0*8 and obviously 4*4 is bigger so (987 654 324)^2 is the larger number.

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  24. To Add On: It's something like finding the common factor of these two numbers and comparing the leftover numbers to find out which is bigger so its easier to compare also and don't require the calculator.

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  25. @Lovy

    (987 654 324)^2 =(987 654 324)(987 654 324)
    =987 654 320^2+4*4 {How did you arrive at this step} Hm... possible to explain algebraically?
    (or are there missing brackets?)

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  26. If 987 654 324=a,
    987 654 320= a-4
    987 654 320=a+4
    So,
    a^2=a^2
    (a-4)*(a+4)=a^2-4
    a^2>a^2-4
    987 654 324^2>(987 654 320)*(987 654 328)

    Ans: The 'Blue' Number is larger.

    Dunno if my working is correct...

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  27. @Jiajun:

    Good attempt :)
    Is there a typoerror for the 3rd line?
    "987 654 320=a+4"
    Next, why "(a-4)*(a+4)=a^2-4"?
    Check the calculation again...

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