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Jump Size Problem
Jump Jackpot
Jumpy and Grumpy play a game.Grumpy places a treasure on some number.
Jumpy and Grumpy play a game.Grumpy places a treasure on some number.
For example, he may place it on 24.
Jumpy chooses a jump size. If he chooses 4, then he has to jump only on multiples of 4, starting at 0.
Jumpy gets the treasure if he lands on the number where Grumpy placed it.
Which jump sizes will get Jumpy to land on 24?
If he chooses 4: Jumpy lands on 4 → 8 → 12 → 16 → 20 → 24 → 28 → ...
Other successful jump sizes are 2, 3, 6, 8 and 12.
Jumpy chooses a jump size. If he chooses 4, then he has to jump only on multiples of 4, starting at 0.
Jumpy gets the treasure if he lands on the number where Grumpy placed it.
Which jump sizes will get Jumpy to land on 24?
If he chooses 4: Jumpy lands on 4 → 8 → 12 → 16 → 20 → 24 → 28 → ...
Other successful jump sizes are 2, 3, 6, 8 and 12.
What about the jump of sizes 1 and 24? Yes, they also will land on 24.
The numbers 1, 2, 3, 4, 6, 8, 12, 24 all divide 24 exactly.
The numbers 1, 2, 3, 4, 6, 8, 12, 24 all divide 24 exactly.
Recall that such numbers are called factors or divisors of 24.
What jump size can reach both \(15\) and \(30\)? There are multiple jump sizes possible. Try to find them all.
Answer:
To find the jump sizes that allow Jumpy to land on both \(15\) and \(30\), we need to find the common factors of both numbers:
Factors of 15: \(1, 3, 5, 15\)
Factors of 30: \(1, 2, 3, 5, 6, 10, 15, 30\)
The common numbers in both lists are \(1, 3, 5,\) and \(15\). Therefore, these are all the possible successful jump sizes.
Grumpy increases the level of the game. Two treasures are kept on two different numbers. Jumpy has to choose a jump size and stick to it. Jumpy gets the treasures only if he lands on both the numbers with the chosen jump size. As before, Jumpy starts at 0.
Grumpy has kept the treasures on 14 and 36. And, Jumpy chooses a jump size of 7.
Will Jumpy land on both the treasures?
Will Jumpy land on both the treasures?
Starting from 0, he jumps to 7 → 14 → 21 → 28 → 35 → 42 ...
We see that he landed on 14 but did not land on 36, so he does not get the treasure.
What jump size should he have chosen?
The factors of 14 are: 1, 2, 7, 14. So, these jump sizes will land on 14.
The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18 and 36. These jump sizes will land on 36.
So, the jump sizes of 1 or 2 will land on both 14 and 36. Notice that 1 and 2 are the common factors of 14 and 36.
The jump sizes using which both the treasures can be reached are the common factors of the two numbers where the treasures are placed.
Question:What jump size can reach both \(15\) and \(30\)? There are multiple jump sizes possible. Try to find them all.
Answer:
To find the jump sizes that allow Jumpy to land on both \(15\) and \(30\), we need to find the common factors of both numbers:
Factors of 15: \(1, 3, 5, 15\)
Factors of 30: \(1, 2, 3, 5, 6, 10, 15, 30\)
The common numbers in both lists are \(1, 3, 5,\) and \(15\). Therefore, these are all the possible successful jump sizes.
Question:
Look at the table below (containing numbers \(31\) to \(70\), where multiples of \(4\) are shaded and multiples of \(6\) are circled).
Is there anything common among the shaded numbers?
Is there anything common among the circled numbers?
Which numbers are both shaded and circled? What are these numbers called?
Answer:
Table Analysis
Yes, all the shaded numbers are multiples of 4.
Yes, all the circled numbers are multiples of 6.
The numbers that are both shaded and circled are \(36, 48,\) and \(60\). These numbers are called common multiples of \(4\) and \(6\).
Figure it Out
1. Find all multiples of \(40\) that lie between \(310\) and \(410\).
Answer:
The multiples of \(40\) in this range are \(320, 360,\) and \(400\).
2. Who am I?
a. I am a number less than 40. One of my factors is 7. The sum of my digits is 8.
Answer:
a. I am a number less than 40. One of my factors is 7. The sum of my digits is 8.
Answer:
Let's look at multiples of \(7\) less than \(40\): \(7, 14, 21, 28, 35\).
Checking the sum of their digits: \(3 + 5 = 8\).
The number is \(35\).
b. I am a number less than 100. Two of my factors are 3 and 5. One of my digits is 1 more than the other.
Answer:
A number having factors \(3\) and \(5\) must be a multiple of \(15\).
Multiples of \(15\) less than \(100\) are: \(15, 30, 45, 60, 75, 90\).
In the number \(45\), the digit \(5\) is exactly \(1\) more than \(4\).
The number is \(45\).
3. Find a perfect number between 1 and 10.
(A perfect number is a number for which the sum of all its factors is equal to twice the number).
(A perfect number is a number for which the sum of all its factors is equal to twice the number).
Answer: Let's test \(6\). The factors of \(6\) are \(1, 2, 3,\) and \(6\).
\[\text{Sum of factors} = 1 + 2 + 3 + 6 = 12\]
Since \(12\) is twice of \(6\) (\(2 \times 6 = 12\)), \(6\) is a perfect number.
4. Find the common factors of:
a. 20 and 28
Factors of 20: \(1, 2, 4, 5, 10, 20\)
Factors of 28: \(1, 2, 4, 7, 14, 28\)
Common Factors: \(1, 2, 4\)
b. 35 and 50
Factors of 35: \(1, 5, 7, 35\)
Factors of 50: \(1, 2, 5, 10, 25, 50\)
Common Factors: \(1, 5\)
c. 4, 8 and 12
Factors of 4: \(1, 2, 4\)
Factors of 8: \(1, 2, 4, 8\)
Factors of 12: \(1, 2, 3, 4, 6, 12\)
Common Factors: \(1, 2, 4\)
Factors of 4: \(1, 2, 4\)
Factors of 8: \(1, 2, 4, 8\)
Factors of 12: \(1, 2, 3, 4, 6, 12\)
Common Factors: \(1, 2, 4\)
d. 5, 15 and 35
Factors of 5: \(1, 5\)
Factors of 15: \(1, 3, 5, 15\)
Factors of 35: \(1, 5, 7, 35\)
Common Factors: \(1, 5\)
Factors of 5: \(1, 5\)
Factors of 15: \(1, 3, 5, 15\)
Factors of 35: \(1, 5, 7, 35\)
Common Factors: \(1, 5\)
5. Find any three numbers that are multiples of 25 but not multiples of 50.
Answer:
Answer:
Multiples of \(25\) are \(25, 50, 75, 100, 125, 150\dots\)
Excluding the multiples of \(50\), three such numbers can be \(25, 75,\) and \(125\).
6. Anshu and his friends play the 'idli-vada' game with two numbers, which are both smaller than 10. The first time anybody says 'idli-vada' is after the number 50. What could the two numbers be which are assigned 'idli' and 'vada'?
Answer:
Saying 'idli-vada' represents a common multiple.
Since it is said for the first time after 50, we need to find a Least Common Multiple (LCM) slightly greater than \(50\) for two single-digit numbers (\(<10\)).
The numbers \(6\) and \(9\) have an LCM of \(54\), which perfectly fits the rule.
7. In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers?
Answer: We need the common factors of \(28\) and \(70\):
Factors of 28: \(1, 2, 4, 7, 14, 28\)
Factors of 70: \(1, 2, 5, 7, 10, 14, 35, 70\)
Common Jump Sizes: \(1, 2, 7,\) and \(14\)
7. In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers?
Answer: We need the common factors of \(28\) and \(70\):
Factors of 28: \(1, 2, 4, 7, 14, 28\)
Factors of 70: \(1, 2, 5, 7, 10, 14, 35, 70\)
Common Jump Sizes: \(1, 2, 7,\) and \(14\)
8. In the diagram below, Guna has erased all the numbers except the common multiples. Find out what those numbers could be and fill in the missing numbers in the empty regions.
(The Venn diagram shows common multiples \(24, 48, 72\) in the center intersection loop)
Answer:
Here is the filled-in Venn diagram, followed by step-by-step prime factorizations for the remaining questions.
The numbers 24, 48, and 72 given in the central overlapping section are common multiples of 6 and 8. Below is the completed diagram filled with their individual multiples in the empty regions:
The numbers in the intersection are multiples of \(24\). This means the two outer categories are Multiples of 6 and Multiples of 8 (or any other pair whose LCM is \(24\), such as \(3\) and \(8\)).
If choosing \(6\) and \(8\), fill the left circle with other multiples of $6$ (e.g., \(6, 12, 18$) and the right circle with other multiples of \(8\) (e.g., \(8, 16, 32\)).
9. Find the smallest number that is a multiple of all the numbers from 1 to 10, except for 7.
Answer:
The numbers in the intersection are multiples of \(24\). This means the two outer categories are Multiples of 6 and Multiples of 8 (or any other pair whose LCM is \(24\), such as \(3\) and \(8\)).
If choosing \(6\) and \(8\), fill the left circle with other multiples of $6$ (e.g., \(6, 12, 18$) and the right circle with other multiples of \(8\) (e.g., \(8, 16, 32\)).
9. Find the smallest number that is a multiple of all the numbers from 1 to 10, except for 7.
Answer:
We find the LCM of \(1, 2, 3, 4, 5, 6, 8, 9,\) and \(10\):
To find the smallest common number (LCM) for all numbers from 1 to 10, we first write out the prime factorizations for each integer:
2 = \(2\)
3 = \(3\)
4 = \(2 \times 2 = 2^2\)
5 = \(5\)
6 = \(2 \times 3\)
8 =\(2 \times 2 \times 2 = 2^3\)
9 = \(3 \times 3 = 3^2\)
10 = \(2 \times 5\)
Take the highest power of each prime number present across the lists (except 7):
Highest power of \(2 \) is $2^3 = 8 \)
Highest power of \(3 \) is $3^2 = 9 \)
Highest power of \(5 \) is $5^1 = 5 \)
\(\text{LCM} = 2^3 \times 3^2 \times 5^1 \)
\(\text{LCM} = 8 \times 9 \times 5 = \mathbf{360} \)
10. Find the smallest number that is a multiple of all the numbers from 1 to 10.To find the smallest common number (LCM) for all numbers from 1 to 10, we first write out the prime factorizations for each integer:
2 = \(2\)
3 = \(3\)
4 = \(2 \times 2 = 2^2\)
5 = \(5\)
6 = \(2 \times 3\)
8 =\(2 \times 2 \times 2 = 2^3\)
9 = \(3 \times 3 = 3^2\)
10 = \(2 \times 5\)
Take the highest power of each prime number present across the lists (except 7):
Highest power of \(2 \) is $2^3 = 8 \)
Highest power of \(3 \) is $3^2 = 9 \)
Highest power of \(5 \) is $5^1 = 5 \)
\(\text{LCM} = 2^3 \times 3^2 \times 5^1 \)
\(\text{LCM} = 8 \times 9 \times 5 = \mathbf{360} \)
Answer:
Highest power of $2$ is \(2^3 = 8\)
Highest power of $3$ is \(3^2 = 9\)
Highest power of $5$ is \(5^1 = 5\)
Highest power of $7$ is \(7^1 = 7\)
\(\text{LCM} = 2^3 \times 3^2 \times 5^1 \times 7^1 \)
\(\text{LCM}(360, 7) = 360 \times 7 = \mathbf{2520} \)
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