Gauss Contest (NMTC PRIMARY 2019 - V and VI Grades) - Stage I- Problems and Solution

Part A

Problem 1

How many glasses of 120 millilitres can you fill from a 3 litre can of juice?
(A) 20
(B) 24
(C) 25
(D) $60 \mathrm{ml}$ is left in the can after filling as many glasses as possible

Problem 2

The sum of $2211+2213+2215+2217+2219+2221+2223+2225+2227+2229$ is
(A) 22200
(B) 22225
(C) 22250
(D) 22275

Problem 3

If a number is first multiplied by $\frac{4}{ 7}$ and then divided by $\frac{12}{7}$ then it is equivalent to which of the following operations on the number?

(A) multiplying by $\frac{1}{3}$
(B) dividing by $\frac{1}{3}$
(C) multiplying by 3
(D) dividing by $\frac{2}{3}$

Problem 4

$X$ is a 5 digit number. Let $Y$ be the sum of the digits of $X$ . Let $Z$ be the sum of the digits of $Y$ . Then the maximum possible value that $Z$ can have is
(A) 9
(B) 8
(C) 10
(D) 12

Problem 5

A square is constructed on a graph paper which has a square grid of $1 \mathrm{~cm}$ width. Ram paints all the squares which cross the two diagonals of the square and finds that there are 19 of them. Then the side of the square is
(A) 20
(B) 19
(C) 10
(D) 9

Problem 6

Look at the set of numbers ${2,3,5,7,8,10,12}$ . Four numbers are selected from this and made into two pairs. The pairs are added and the resulting two numbers are multiplied. The smallest such product is
(A) 72
(B) 60
(C) 54
(D) 64

Problem 7

Anita wants to enter a number into each small triangular cell of the triangular table. The sum of the numbers in any two such cells with a common side must be the same. She has already entered two numbers. What is the sum of all the numbers in the table?

(A) 19
(B) 20
(C) 21
(D) 22

Problem 8

Where A, B, C, D, E are distinct digits satisfying this addition fact, then E is

(A) 3
(B) 5
(C) 2
(D) 4

Problem 9

A large rectangle is made up of eleven identical rectangles whose longer sides are $21 \mathrm{~cm}$ long. The perimeter of the large rectangle in $\mathrm{cm}$ is

(A) 150
(B) 126
(C) 108
(D) 96

Problem 10

Sum of the odd numbers from 1 to 2019 both inclusive, is divisible by
(A) only 100
(B) only 101
(C) both 100 and 101
(D) neither by 100 nor by 101

Problem 11

The circumference of a circle is numerically greater than the area of the circle. Then the maximum length of the radius cannot be greater than
(A) 2
(B) $\frac{7}{4}$
(C) $\frac{9}{5}$
(D) $\frac{11}{6}$

Problem 12

A calendar for 2019 is made using 4 sheets, each sheet having 3 months. The total number of days shown in each of the four sheets $\left(1^{\text {st }}, 2^{\text {nd }}, 3^{\text {red }}, 4^{\text {th }}\right)$ respectively is
(A) $(90,91,92,92)$
(B) $(90,92,91,92)$
(C) $(90,92,91,92)$
(D) $(90,92,92,91)$

Problem 13

Triples of odd numbers $(a, b, c)$ with $a<b<c$ , with $a, b, c$ from 1 to 10 are generated such that $a+b+c$ is prime number. The number of such triple is
(A) 5
(B) 6
(C) 7
(D) 3

Problem 14

A number leaves a remainder 2 when divided by 6 . Then the possible remainder(s) when the same number is divided by 9 is
(A) ${1,4,7}$
(B) ${2,5,8}$
(C) ${5,8}$
(D) ${2,5}$

Problem 15

A box of dimension $40 \times 35 \times 28$ units is used to keep smaller cuboidal boxes so that no space is left between the boxes. If the box is packed with 100 such smaller boxes of the same size, then dimension of the smaller box is
(A) $7 \times 8 \times 7$
(B) $8 \times 7 \times 7$
(C) $7 \times 7 \times 8$
(D) $20 \times 7 \times 7$

Part B

Problem 16

The number of two digit numbers which are divisible by the sum of their digits is $\rule{1cm}{0.15mm}$ .

Problem 17

Given below is the triangular form of AMTI.

The number of ways you can spell AMTI, top to bottom, right to left or left to right or a combination of these is $\rule{1cm}{0.15mm}$ .

Problem 18

The number of odd prime numbers less than 100 which can be written as the sum of two squares is $\rule{1cm}{0.15mm}$ .

Problem 19

If $4921 \times D=A B B B D$ then $B$ is $\rule{1cm}{0.15mm}$ .

Problem 20

36 children took a math talent test. For a contestant Anu the number of students who scored above her was 1.5 times the number who scored below her. Her rank when the scores are put in decreasing order is $\rule{1cm}{0.15mm}$ .

Problem 21

Small rectangular sheets of length $\frac{2}{3}$ units and breadth $\frac{3}{5}$ units are available. These sheets are assembled and pasted in a big cardboard sheet, edge to edge and made into a square. The minimum number of such sheet required is $\rule{1cm}{0.15mm}$ .

Problem 22

Ramanujan's number is 1729. The number of composite divisors of 1729 less than 1729 is $\rule{1cm}{0.15mm}$ .

Problem 23

$\mathrm{N}$ is an 100000 digit number with no zero digit and the sum of the digits of $\mathrm{N}$ is 100001 , then the number of such N's is $\rule{1cm}{0.15mm}$ .

Problem 24

Peter 8 years old asked his mother how old she was. She said, "when you are as old as I am now, I will be 54 years old". Peter's mother's current age is $\rule{1cm}{0.15mm}$ .

Problem 25

A string of beads has a recurring pattern as follows : 5 blue, 4 black, 4 white, 5 blue, 4 black, 4 white and so on. The colour of the $321^{st}$ bead is $\rule{1cm}{0.15mm}$ .