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FOURTH GRADE / MATHEMATICS / NATURAL NUMBERS 1.1

MATHEMATICS IS EASY. PROVIDED THAT ONE WORKS ENOUGH .
ENOUGH IS SKILL DEPENDENT, HAVE TO FIND "ENOUGH" FOR YOUR SIZE.

 In mathematics, natural numbers are

    ♦ 
the ordinary counting numbers 1, 2, 3, ...

    ♦ sometimes zero is also included.


    Ref.-1

    Ref.-2
y
1. "0" being not considered as a natural number, which one of the followings is the least 4-digit natural number ?

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RY1
RY2



a) 0100
b) 1111
c) 1000
d) 1110


"0" is not counted as a natural number, but it should be used as a digit.
y
2. Which choice gives the sum of all 2-digit natural numbers having 3 as the second digit ?

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RY1 RY2



a) 355
b) 345
c) 255
d) 305
30+30+1++30+2+...+30+9 = 10x30+1+2+3+...+9.
3. Which natural number N has the property that 3xN is equal to the sum of the natural numbers less than N ?

a) 7
b) 6
c) 8
d) 9
1, 2, ... ,6 < 7 ; 1+2+3+4+5+6=21 .
4. How can we show that there exists a unique natural number N such that 3xN = Σ where Σ is the sum of all natural numbers less than or equal to N ?

a) Existence can be shown by trial, 5 is pretty easy to find. For uniqueness we have to show that the equality forces inequality for any M > N or M < N
b) Existence can be shown by trial, 5 is pretty easy to find. For uniqueness we have to show that the equality forces inequality for any M > N.
c) Existence can be shown by trial, 5 is pretty easy to find.
d) Existence can be shown by trial, 5 is pretty easy to find. For uniqueness we have to show that the equality forces inequality for any M < N.
The answer to the question is clear. The aim is to adapt you to mathematical thinking.
5. Which choice gives the sum of maximum and minimum natural numbers having at most 4-digit with 3 at hundreds and 7 at tens places ?

a) 10749
b) 9749
c) 370
d) 1370
Numbers should be something like X37Y where X and Y are any one of the digits. For max. X=Y=9 and for min X does not exist at all due to "at most" and Y=0
6. If the number Δ2 having the digit represented by Δ at tens and 2 at units places is equal to 12xΔ, which one of the following digits may be the right choice for Δ ?

a) 2
b) 1
c) 0
d) 3
The equation 10xΔ + 2 = 12xΔ may be solved for Δ. Trial and error works also.
7. Which of the digits has the greatest place value in 27386 ?

a) 8
b) 7
c) 6
d) 2
Units, tens, hundreds, ... .
y
8. Which is the correct way of writing the natural number five hundred and thirty-four thousand, three hundred and seventy-six?

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RY1 RY2



a) 534,376
b) 299,534
c) 534,736
d) 534,637
Click on ry1 if needed .
9. Which of the following natural numbers is always an even number ?

a) Sum of 2 integers.
b) Multiplication of 2 integers.
c) Twice an integer.
d) Difference of 2 inttegers.


A trivial case ... .
10. Which operation defined on natural numbers forces us to extend the set of natural numbers ?

a) multiplication and division
b) summation and subtraction
c) subtraction only
d) subtraction and division
inverse operations .
11. What is the difference of the minimal 4-digit number and the maximal 3-digit number if both are restricted to use a digit only once ?

a) 1
b) 23
c) 24
d) 26
The max. is 987 within restriction .
12. How many times the digit 0 should be used in writing the natural numbers from one up to hundred and eleven ?

a) 11
b) 12
c) 9
d) 21
Remember one-o-one, one-o-two, ... etc. .
13. Which of the followings is the correct way of writing the natural number 300,047 ?

a) Threehundert thousand fourty seven
b) Three hundred thousand, fourty-seven
c) Threehundredthousand, fourty-seven
d) Three hundred thousand, fourty seven
Separate the words in order to emphasize them and their place values + ry1 of question 8.
14. The natural number 300,047 consists of ... ?

a) Three times hudred thousand, four times ten and a seven
b) Three times ten thousand and four times ten
c) Three times ten thousand, four times ten and a seven
d) Three times ten thousand, four times hudred and a seven
Units (ones), tens , hundreds, thousands, ten thousands, ... etc. ?
15. Which natural number should follow in the sequence 7, 15, 12, 20, 17, ... ?

a) 25
b) 17
c) 27
d) 19
plus 8, minus 3 .
16. Which natural number should follow in the sequence 2, 5, 26, ... ?

a) 677
b) 33
c) 337
d) 435
Square (multiply with itself) and add 1 .
17. If the word "OTTA" is encrypted as 3557, the word "KRAK" is supposed to be encrypted as ... ?

!


a) 6946
b) 9469
c) 8378
d) 4934
A correponds to 7
18. How many 4-digit natural numbers can be constructed with the digits of the natural number 50,079 under the restriction that every digit can only be used once ?

!


a) 9
b) 12
c) 18
d) 24
The image is explaining ...
19. In a bag there are 90 pebbles each having a different natural number from 1 to 90 on them.
A player has 10 cards at hand, first card containing the digits 1 to 9 on it, second having the numbers 10 to 18, third having 19 to 27 , ... , and tenth card having 82 to 90.
Drawing a pebble from the bag, which of the followings is the easiest way to determine on which card is the number on the pebble is to be found ?

!


a) To subtract enough number of nines from the number on the pebble.
b) To subtract max. possible number of nines from the number on the pebble.
c) To divide the number on the pebble by nine, ignore the remainder, increase the quotient by 1
d) To divide the number on the pebble by nine.
Look at the figure and try .
20. In a bag there are 90 pebbles each having a different natural number from 1 to 90 on them.
A player has 18 cards at hand, first card containing the numbers 1, 19, 37, 55 and 73 on it, second having the consecutives of the numbers on the first, third the consecutives of the 2nd and so on as seen on the figure below.
Drawing a pebble from the bag, which of the followings is the easiest way to determine on which card is the number on the pebble is to be found ?

!


a) To sum up the digits on the pebble.
b) To sum up the digits till one digit number and add up a 9 if necessary.
c) To sum up the digits on the pebble and add up a 9.
d) To divide the number by 18 and get the remainder.
Look at the figure and try .
21. Which is the greater one of the two natural numbers having a sum of 87 and a difference of 37 ?

a) 62
b) 63
c) 60
d) 61
A+B = 87 and A-B = 37 should be solved.
Trial and error should be preferred. If A=61 then B must be 26 according to the first, sum, equation. 61-26=35.
Difference must be greater.
If A is 63 B is 24 and the difference is 39.
y
22. Interchanging the digits of a 2-digit number and subtracting the smaller 2-digit from the bigger 2-digit results with a 4 at the units of the difference.
Which of the followings should be the difference ?


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RY1 RY2



a) 24
b) 34
c) 44
d) 54
The digits of the 2-digit number can not be the same, otherwise no diff. at all.
23. The inequality A > 85/17 is given.
Which of the followings is the minimal of all possible A's ?

a) 5
b) 6
c) 4
d) 3
85/17 = 5 . That means A is supposed to be a natural number greater than 5.
24. In case of a 3-digit number with all digits differing the digit at ones and hundreds are interchanged and the difference (positive) is taken.
If the units (digit) of the difference is 3, which of the followings should be the difference ?

a) 393
b) 933
c) 693
d) 396
Originally ABC happens to be CBA after interchanging.
ABC = 100xA + 10xB + C
CBA = 100xC + 10xB + A

Difference = 100x(A-C) + C - A = 99 x (A - C) i.e. a multiple of 99.
It should also be a multiple of 9 .
For the difference to be a natural number (positive) is the necessary condition A>C
Since the middle digit is the same for both, the middle digit of the diff. sholud be 9.
25. Both of the equalities 2x8 = 16 and 4x4 = 16 are true.
How can we show that these equalities give axactly the same 16 and all sixteens are exactly the same using samples around us ?

a) Easy task, trivial.
b) Get 32 beans and make a 4 by 4 group and a 2 by 8 group.
c) Mathematical quantities are conceptual extensions from our everyday experience, it is not physically possible to show their exact equality without error.
d) Use extremely sensitive scales.
.
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