Math Level: Prime Numbers

Grade Level: middle school

Topics: Prime numbers, square roots, factors, division

The Question:

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Is 613 a prime number?

The Answer:

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Yes it is prime.

You need only check odd numbers up to the closest square root of 613, namely 24.

Therefore – we will check – 3,5,7,9,11,13,15,17,19,21,23.

We can eliminate 3 because 6 + 1 + 3 = 10, the sum is not divisible by 3. Therefore multiples of 3, namely 9, 15 and 21 cannot divide 613.

We can eliminate 5 because 613 does not end in a 0 or 5.

We can eliminate 11 because the of the way 11 works it would mean 613 is 63 x 11 which would be 693.

Therefore we have to check 13, 17, 19 and 23. Using division, none of them are factors of 613.

Therefore, 613 is prime!

If it wasn’t – how many different divrei Torah would be composed trying to associate different miztvos with different factors. 613 is prime and therefore all miztvos are equal

(24)

]]>In parsha Nasso (Bamidbar 7:3) the Nissim (tribal leaders) donated carts to carry the parts of the Mishkan. The Gemara in Shabbos 98a discusses the definition of a public domain. The Gemara specifically addresses whether a roofed area can be a public domain. They bring the example of the area under the carts used to carry the planks of the walls of the Mishkan. The Gemara states that area under the carts was public domain, however, not all of the area under the carts were covered. The question becomes if the public domain was the area under the carts that was covered or is the public domain only the uncovered space under the cart. As part of this discussion the Gemara investigates if there was any significant space under the carts that was not covered.

The frame of the cart did not have a bottom and the planks were laid across the sides of the frame. The length of the carts were 5 amahs long. (See Figure A) the planks of the Mishkan were 1 amah on one side and 1.5 amah on the other and 10 amah long. (See Figure B)

There are 6 tefach to each amah. If there is less than 3 tefach of open space that amount of space is considered insignificant (i.e. there is no halacially open space under the cart)

Option 1:

The planks are laid out on their wider side of 1.5 amah, the maximum number of planks are laid out across the cart without going over the side and they are evenly spaced. What is the maximum number of planks that can be laid across the cart? How much open space would be between each plank? Is this amount of open space halacially significant?

Answer (See Figure C)

a = 5 amah = the length of the cart

b = 1.5 amah = the width of the plank

4 x 1.5 amah = 6, so that is too many to fit.

3 x 1.5 amah = 4.5, so that fits with .5 amah left over for two spaces

C= .5 / 2 = .25 amah = 6 tefach * .25 = 1.25 tefach

Since 1.25 tefach is less than 3 the open space is not halacially significant.

Option 2:

The planks are laid out on the narrower side of 1 amah, the maximum number of planks are laid out across the cart without going over the side, there is at least some space between each plank and they are evenly spaced. What is the maximum number of planks that can be laid across the cart? How much open space would be between each plank? Is this amount of open space halacially significant?

Answer (See Figure D)

a = 5 amah = the length of the cart

b = 1 amah = the width of the plank

5 x 1 amah = 5, so there would be no space between the planks.

4 x 1 amah = 4, so that fits with 1 amah left over for three spaces

C= 1 / 3 amah = 6 tefach * 1/3 = 2 tefach

Since 2 tefach is less than 3 the open space is not halacially significant.

Option 3:

The Gemara offers another possibility that four planks were clustered with two on each side of the cart with all the open space in the middle. The planks are laid out on the narrower side of 1 amah. How much open space would in the middle? Is this amount of open space halacially significant?

Answer (See Figure E)

a = 5 amah = the length of the cart

b = 1 amah = the width of the plank

5 x 1 amah = 5, so there would be no space between the planks.

4 x 1 amah = 4, so that fits with 1 amah left over between the two planks on either side.

C= 1 amah = 6 tefach

Since 6 tefach is more than 3 the open space is halacially significant.

(104)

]]>We know that the law against carrying on Shabbos derives from the carrying of the aron in the wilderness. A proof can be made that the aron was “carried” even though its bottom was lifted over 10 tefach from the ground. Since this was still carrying then it is a proof that carrying an item over 10 tefach high does not create a private domain.

The aron was 10 tefach in height. (See a) The rings for the poles used to carry the aron were 1/3 of the way down the side. (See b) The shoulder height of the average Israelite was 18 tefach. (See c) Using this information show how high off the ground was the aron when it was being carried. (Solve for d)?

c – 2/3a = d

18 – 2/3 * 10 = 11.33

(97)

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Here are some examples of how this works:

*Figure 1*

If there is a row of carrots and a row of beets parallel to each other then the rows need to be at least 3 tefach apart.

*Figure 2*

If there is a row of carrots and a row of beets perpendicular to each other and one row is near the side of another row then the closest seed of each seed has to be at least 3 tefach apart.

*Figure 3*

The exception to the rule in *figure 2* is when the two lines meet at a corner. If the two lines are perpendicular to each other and they ONLY meet at a corer then they do NOT need to be 3 tefach apart.

*Figure 1*

*Figure 2*

*Figure 3*

How can you fit the following in to a field that is in a 6 x 6 tefach square?

A row of parsnips 5.9 tefach long

A row of carrots 5.9 tefach long

A row of beets 5.9 tefach long

A row of onions 5.9 tefach long and

One potato

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18 x 3/4 = 13.5 minutes according to Rabbah

18 x 2/3 = 12 minutes according to Rav Yosef

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The formula for exponential decay is

y = a (1 – r)^{x}

y = current eyesight

a = original amount of eyesight (in this case 100%)

r= rate of decay (in this case 1/500)

x=number of steps (in this question 500)

y= 100 (1- 1/500)^{500}

y= 100 (1-.002)^{500}

y= 100 (.998)^{500}

y= 100 * .3675112548

y = 36.75112548% of vision remaining

(243)

]]>** **

3 x 3 = 9

24 – 9 = 15 hours

(176)

]]>If a day begins at 6:50am and ends at 7:02pm how long is each HH between 6:50am and 7:02pm?

6:50am to 7:02pm = 12 hours and 12 minutes in SH

12 hours x 60 minutes = 720 minutes, 720 minutes + 12 minutes = 732 minutes

732 / 12 = 61 minutes per HH.

There are two time slots which are discussed during which one can daven Mincha. One is a larger time slot and one is a smaller time slot. The larger time slot starts 6.5 halachic hours (HH) into the day and ends at the end of the day. The smaller time slot begins 9.5 halachic hours (HH) into the day and ends at the end of the day.

What are the times for the long and short periods for Mincha if a day begins at 6:50am and ends at 7:02pm?

Long Mincha

How many minutes in 6.5 HH?

6.5 HH x 61 minutes = 396.5 minutes

396.5 / 60 = 6 hours and 36.5 minutes

6 hours and 36.5 minutes after 6:50am is 1:26:30pm

The long Mincha is from 1:26:30pm to 7:02pm

Short Mincha

How many minutes in 9.5 HH?

9.5 HH x 61 minutes = 579.5 minutes

579.5 / 60 = 9 hours and 39.5 minutes

9 hours and 39.5 minutes after 6:50am is 4:29:30pm

The short Mincha is from 4:29:30pm to 7:02pm

The opinion of Rebbe Yehudah in the Gemara is that Mincha can only be said until Plag HaMincha. Plag HaMincha is the midpoint between the beginning and the end of the short Mincha.

If a day begins at 6:50am and ends at 7:02pm when can you daven Mincha according to Reb Yehudah?

The short Mincha is from 4:29:30pm to 7:02pm

4:29:30pm to 7:02pm = 2 hours 32.5 minutes = 152.5 minutes

152.5 / 2 = 75.25 minutes

4:29:30pm + 72.25 minutes = 5:41:45pm

A standard hour (SH) that we normally use is 60 minutes long. A halachic hour (HH) is 1/12 of the time between the start of the day and the end. On a day where there is an equal length of day and night each halachic hour would be 60 minutes, but it is a different length all other days. (Berachos 26b)

Assuming you are in Anchorage Alaska in May and the If a day begins at 5:24am and ends at 10:24 pm how long is each HH between 5:24am and 10:24pm?

5:24am to 10:24pm= 18(SH) hours

18(SH) hours x 60 minutes = 1,080 minutes

1080 / 12 = 90 minutes per HH.

There are two time slots which are discussed during which one can daven Mincha. One is a larger time slot and one is a smaller time slot. The larger time slot starts 6.5 halachic hours (HH) into the day and ends at the end of the day. The smaller time slot begins 9.5 halachic hours (HH) into the day and ends at the end of the day.

What are the times for the long and short periods for Mincha if a day begins at 5:24am and ends at 10:24 pm?

Long Mincha

How many minutes in 6.5 HH?

6.5 HH x 90 minutes = 585 minutes

585 / 60 = 9.75 hours

9.75 after 5:24am is 3:09pm

The long Mincha is from 3:09pm to 10:24pm

Short Mincha

How many minutes in 9.5 HH?

9.5 HH x 90 minutes = 855 minutes

855 / 60 = 14.25 hours

14.25 hours after 5:24am is 7:39pm

The short Mincha is from 7:39pm to 10:24pm

The opinion of Rebbe Yehudah in the Gemara is that Mincha can only be said until Plag HaMincha. Plag HaMincha is the midpoint between the beginning and the end of the short Mincha.

If a day begins at 5:24am and ends at 10:24 pm when can you daven Mincha according to Reb Yehudah?

The short Mincha is from 7:39pm to 10:24pm

7:39pm to 10:24pm= 2 hours 45 minutes = 165 minutes

165/ 2 = 87.5 minutes

7:39pm + 87.5 minutes = 9:06:30 pm

(237)

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What fraction of a sha’ah is a rega?

1/24 x 1/24 x 1/24 = 1/13,824 of a sha’ah

If a sha’ah is 60 minutes what fraction of a second is a rega?

Number of seconds in a hour = 60 x 60 = 3,600

Number of rega’im in an hour = 13,824

3,600 / 13,824 = 25/96 of a second

(238)

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Author: S. B. Schramm

Math Level: Geometry

Grade Level: 10th-11th

Topics: Chanukah candles

The Question:

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Yehudit is checking to make sure that her candles are between the recommended 3 and 10 tfachim above the ground. Standing 10 amot away she looks down at her candles at an angle of 9°. If her eye height is 3 amot, are her candles at the correct height? (Note: 6 tfachim is equivalent to 1 amah)

According to the Chazon Ish 1 tefach is equivalent to 9.6 cm. What is the height of Yehudit’s candles?

According The Chatam Sofer, 1 tefach is approximately 4.25 inches. What is the height of Yehudit’s candles?

Reuven is walking along when he spots a chanukia in the second story window and wonders if it is below the maximum height of 20 amot above the ground. When he is 50 amot away from the building he looks up to the candles at an 18⁰ angle and down to the base of the building at a 3⁰ angle. How high are the candles? Are they below the maximum?

The Answer:

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Let h be the height of the chanukiah and x be the distance from Yehudit’s eyes down to the candles.

x/10=tan9° so x=10 ∙tan9°=1.58 amot

h=eye height-x=3-1.58=1.42 amot

(6 tfachim)/(1 amah)= h/(1.42 amot)

h=6 ∙1.42=8.52 tfachim

Her chanukiah is at good height.

(9.6 cm)/(1 tefach)= h/(8.52 tfachim)

h=9.6 ∙8.52=81.79 cm

(4.25 inches)/(1 tefach)= h/(8.52 tfachim)

h=4.25∙8.52=36.21 inches

Let a be the distance from Reuven’s eyes up to the candles and b be the distance to the base of the building. Thus

h=a+b

a/50=tan18° so a=50∙tan18°=16.24 amot

b/50=tan3° so b=50∙tan3°=2.62 amot

height of chanukiah is 16.24+2.62=18.86 amot

The chanukiah is below the maximum.

(210)

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